Embeddings of manifolds with boundary: classification

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1 Introduction

In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In $\S$${{Authors|Mikhail Fedorov|Garaev Timur}} {{Stub}} == Introduction == <wikitex>; \label{sec::intro} In this page we present results on embeddings of manifolds with non-empty boundary into Euclidean space. In $\S$\ref{sec::linking_form} we introduce an invariant of embedding of a $n$-manifold in $(n-1)$-space for even $n$. In $\S$\ref{sec::generalisations} which is independent from $\S$\ref{sec::example}, $\S$\ref{sec::linking_form} and $\S$\ref{sec::classification} we state generalisations of theorems from $\S$\ref{sec::general_theorems} to highly-connected manifolds. For a [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Introduction|general introduction to embeddings]] as well as the [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation and conventions|notation and conventions]] used on this page, we refer to \cite[$\S\S$5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\alpha\partial$] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

Then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2]. Part (b) in the case $n>2$$n>2$ is proved in [Edwards1968, $\S$$\S$ 4, Corollary 5]. The case $n=1$$n=1$ is clear. Case $n=2$$n=2$ can be proved using the ideas presented below.

Both parts of Theorem 2.1 are special cases of Theorem 7.2.

The inequality in part (b) is sharp, see Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. Since $2(n-1) < 2n-1$$2(n-1) < 2n-1$, by general position we may assume that $g|_{X}$$g|_{X}$ is an embedding. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Alternative proof of Theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by Lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $N$$N$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial N$$\phi : \partial D^i \times D^{n-i} \to \partial N$, $f : N \to \mathbb R^m$$f : N \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(N)$$f(N)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ whose the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $i-1<2n-1 - (n-i)$$i-1<2n-1 - (n-i)$, it follows that $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$ and there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).

Lemma 3.4. Assume that $U$$U$ is a compact $n$$n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$$\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$$i\leq n-1$, $f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$$f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$ are embeddings and $F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$$F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$ is a concordance between $f_0|_U$$f_0|_U$ and $f_1|_U$$f_1|_U$.

$(a)$$(a)$ If also that $m\geq 2n$$m\geq 2n$, then there is an extension of $F$$F$ to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

$(b)$$(b)$ Assume also that there is a concordance $G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$$G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$ between $f_0|_{D^i\times 0}$$f_0|_{D^i\times 0}$ and $f_1|_{D^i\times 0}$$f_1|_{D^i\times 0}$, and on $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$ there is a field of $n-i$$n-i$ linear independed normal vectors whose restrictions to $G(\partial D^i\times 0\times [0, 1])$$G(\partial D^i\times 0\times [0, 1])$, to $G(D^i\times 0\times 0)$$G(D^i\times 0\times 0)$, and to $G(D^i\times 0\times 1)$$G(D^i\times 0\times 1)$ are tangent to

$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$ $$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$

respectively. Then $F\cup G$$F\cup G$ extends to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

Proof of the Theorem 2.2 using Lemma 3.1 and Lemma 3.4.(a). Denote by $f_0, f_1$$f_0, f_1$ any two embeddings of $N_0$$N_0$ into $\mathbb{R}^m$$\mathbb{R}^m$. In the following paragraph we show that there is a concordance between $f_0$$f_0$ and $f_1$$f_1$. From Implies Isotopy Theorem it follows that there is an isotopy between $f_0$$f_0$ and $f_1$$f_1$. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps of indices at most $n-1$$n-1$. Denote by $U^l$$U^l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching the first $l$$l$ handles, starting with $U^1\cong D^n$$U^1\cong D^n$. Define a concordance $F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$ recursively. Take any concordance $F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$ between $f_0|_{U^1}$$f_0|_{U^1}$ and $f_1|_{U^1}$$f_1|_{U^1}$. Let us define a concordance $F_l$$F_l$ between $f_0|_{U^l}$$f_0|_{U^l}$ and $f_1|_{U^l}$$f_1|_{U^l}$ using a concordance $F_{l-1}$$F_{l-1}$ of $U^{l-1}$$U^{l-1}$ between $f_0|_{U^{l-1}}$$f_0|_{U^{l-1}}$ and $f_1|_{U^{l-1}}$$f_1|_{U^{l-1}}$. Denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$$\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$ the $l$$l$-th attaching map. Since $i\leq n-1$$i\leq n-1$, by Lemma 3.4.(a) it follows that there is an extension of $F_{l-1}$$F_{l-1}$ to a concordance

$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$ $$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$$

between the restriction of $f_0$$f_0$ and $f_1$$f_1$ to $U^{l-1}\cup_\phi D^i\times D^{n-i}$$U^{l-1}\cup_\phi D^i\times D^{n-i}$.

$\square$$\square$

Proof of Lemma 3.4.(a) using Lemma 3.4.(b). In the following text we identify $D^i\times D^{n-i}\times [0, 1]$$D^i\times D^{n-i}\times [0, 1]$ and $D^i\times [0, 1]\times D^{n-i}$$D^i\times [0, 1]\times D^{n-i}$. Define map

$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$ $$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$$

by the formula:

Tex syntax error

$$\displaystyle \bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \\ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right.$$

Since

$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$ $$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$$

by general position there is an embedding

$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$ $$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$$

such that $F\bar{\phi} = G$$F\bar{\phi} = G$ on $\partial (D^i \times 0\times [0, 1])$$\partial (D^i \times 0\times [0, 1])$ and $F(\mbox{Int} (U\times [0, 1]))$$F(\mbox{Int} (U\times [0, 1]))$ has a finite number of intersection points with $G(\mbox{Int}( D^i\times 0\times [0, 1]))$$G(\mbox{Int}( D^i\times 0\times [0, 1]))$. Then by an isotopy $G_t$$G_t$, where $G_0=G$$G_0=G$, fixed on $\partial (D^i\times 0\times [0, 1])$$\partial (D^i\times 0\times [0, 1])$ we can "push out" the self-intersection points toward $F(\partial (U\times [0, 1]))$$F(\partial (U\times [0, 1]))$ so that $G_1(D^i\times 0\times [0, 1])$$G_1(D^i\times 0\times [0, 1])$ does not intersect $G(U\times [0, 1])$$G(U\times [0, 1])$. Then we may assume that $F\cup G$$F\cup G$ is an concordance between the restrictions of $f_0$$f_0$ and $f_1$$f_1$ on $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$$U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$.

Denote by $A$$A$ the $(n-i)\times (n+1)$$(n-i)\times (n+1)$ matrix whose rightmost $(n-i)\times (n-i)$$(n-i)\times (n-i)$ submatrix is the identity matrix, and whose other elements are zeroes. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$$\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$ whose $k$$k$-th vector has coordinates equal to the $k$$k$-th row in $A$$A$. Then $d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$$d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$ is the vector field tangent to $\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$$\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$. For $x\in G(\partial (D^i\times 0\times [0, 1])$$x\in G(\partial (D^i\times 0\times [0, 1])$ denote by $v'$$v'$ the projection of $d(F\cup f_0\cup f_1)d\phi (v(x))$$d(F\cup f_0\cup f_1)d\phi (v(x))$ to the intersection of normal space to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$, and tangent space to $F(\partial(U\times [0, 1]))$$F(\partial(U\times [0, 1]))$. Since $i<m+1-(n-i)$$i<m+1-(n-i)$, it follows that $\pi_{i}(V_{m+1, n-i})=0$$\pi_{i}(V_{m+1, n-i})=0$. Hence there is an extension of $v'$$v'$ to a linear independent field of vectors normal to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$. Then by Lemma 3.4.(b) there is an extension of $F\cup G$$F\cup G$ to a concordance $(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$$(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$.

(a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. (To prove part (a) it is sufficient to take as $h$$h$ the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero.) Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. There exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

Definition 5.2. Denote

$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$ $$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$ $$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

Proof of Lemma 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 5.3 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$$\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$$2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$$f(N_0)$ are linearly dependent.

Lemma 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 5.4.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$ $$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 6.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 6.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 6.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$ $$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 6.4. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$ $$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$$

is one-to-one.

Lemma 6.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$ $$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

10 References

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, $\S]{Skopenkov2016c}. In those pages mostly results for closed manifolds are stated. If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$ the linking coefficient \cite[$\S$ 77]{Seifert&Threlfall1980} of two disjoint cycles. We state the simplest results. These results can be deduced from [[Some_calculations_involving_configuration_spaces_of_distinct_points|the Haefliger-Weber deleted product criterion]] \cite[$\S$ 5]{Skopenkov2006}, see \cite[6.4]{Haefliger1963}, \cite[Theorem 1.1$\alpha\partial$]{Skopenkov2002} for the DIFF case and \cite[Theorem 1.3$\alpha\partial$]{Skopenkov2002} for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion. We do not claim the references we give are references to original proofs. == Embedding and unknotting theorems == ; \label{sec::general_theorems} {{beginthm|Theorem}}\label{thm::embed} Assume that $N$ is a compact connected $n$-manifold. (a) Then $N$ embeds into $\mathbb R^{2n}$. (b) If $N$ has non-empty boundary, then $N$ embeds into $\mathbb R^{2n-1}$. {{endthm}} Part (a) is well-known [[Wikipedia:Whitney_embedding_theorem|strong Whitney embedding theorem]]. {{beginthm|Theorem}}\label{thm::unknotting} Assume that $N$ is a compact connected $n$-manifold and either (a) $m \ge 2n+1 \ge 5$ or (b) $N$ has non-empty boundary and $m\geq 2n$. Then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. {{endthm}} Part (a) is Whitney-Wu Unknotting Theorem, see \cite[$\S$ 2, Theorems 2.1, 2.2]{Skopenkov2016c}. Part (b) in the case $n>2$ is proved in \cite[$\S$ 4, Corollary 5]{Edwards1968}. ?he case $n=1$ is clear. Case $n=2$ can be proved using the ideas presenеed below. Both parts of Еheorem \ref{thm::embed} are special cases of the Theorem \ref{thm::k_connect_unknot}. The inequality in part (b) is sharp, see Proposition \ref{exm::linked_boundary}. These basic results can be generalized to highly-connected manifolds (see $\S$\ref{sec::generalisations}). All stated theorems of $\S$\ref{sec::general_theorems} and $\S$\ref{sec::generalisations} for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas. == Proofs of Theorem \ref{thm::embed}.b and Theorem \ref{thm::unknotting}.b == ; \label{sec::pt} The first proof of Theorem \ref{thm::embed}.b uses immersions, while the second does not.

'''First proof of Theorem \ref{thm::embed}.b.''' By the [[Wikipedia:Whitney_immersion_theorem|strong Whitney immersion theorem]] there exist an immersion $g\colon N\to\mathbb R^{2n-1}$. Since $N$ is connected and has non-empty boundary, it follows that $N$ collapses to an $(n-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. Since (n-1) < 2n-1$, by general position we may assume that $g|_{X}$ is an embedding. Since $g$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. The composition $g\circ h$ is an embedding of $N$.{{endproof}} This proof is essentially contained in \cite[Theorem 4.6]{Hirsch1961a} for the Diff case and in references for Theorem \ref{thm::k_connect_embeds} below or in \cite[Theorem 5.2]{Horvatic1971} for the PL case. For the alternative proof we need some lemmas. {{beginthm|Lemma}}\label{p:handle-decompose} \cite{Wall1966} Assume that $N$ is a closed connected smooth $n$-manifold. Then $N_0$ have handle decomposition with indices of attaching map at most $n-1$. {{endthm}} {{beginthm|Lemma}}\label{p:handle-add} Assume that $N$ is a closed smooth $n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i'''Alternative proof of Theorem \ref{thm::embed}.b.''' By Lemma \ref{p:handle-decompose} there is a handle decomposition of $N_0$ with attaching maps $\phi_1,\ldots,\phi_s$ of indices smaller than $n$. Denote by $U_l$ the manifold obtained from $\emptyset$ by the attaching first $l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$ of $U_l$ using an embedding $F_{l-1}$ of $U_{l-1}$. Since the index $i$ of $\phi_l$ is smaller than $n$, by Lemma \ref{p:handle-add} there is extension of $F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$. {{endproof}} {{beginthm|Lemma}}\label{p:handle-field} Let $N$ be a closed smooth $n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial N$, $f : N \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$ on $\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$ there is a field of $n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$ is tangent to $f(N)$. Then $f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$. {{endthm}}

'''Proof of Lemma \ref{p:handle-add} using Lemma \ref{p:handle-field}.''' Since i+1\leq 2n-1$, it follows that there is an extension of $f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$ the $(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$ the field of $n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$ whose the $k$-th vector have coordinates equal to the $k$-th row in $V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$. Denote by $v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$ and tangent space to $\partial N$. Since $i-1<2n-1 - (n-i)$, it follows that $\pi_{i-1}(V_{2n-1, n-i})=0$ and there is an extension of $v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$. Then by Lemma \ref{p:handle-field} there is an extension of $f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$. {{endproof}} In the proof of Theorem \ref{thm::unknotting} we will use Lemma \ref{p:handle-decompose} and Lemma \ref{p:handle-add2}.(a). {{beginthm|Lemma}}\label{p:handle-add2} Assume that $U$ is a compact $n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$, $f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$ are embeddings and $F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$ is a concordance between $f_0|_U$ and $f_1|_U$. $(a)$ If also that $m\geq 2n$, then there is an extension of $F$ to a concordance between $f_0$ and $f_1$. $(b)$ Assume also that there is a concordance $G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$ between $f_0|_{D^i\times 0}$ and $f_1|_{D^i\times 0}$, and on $G(D^i\times 0\times [0, 1])$ there is a field of $n-i$ linear independed normal vectors whose restrictions to $G(\partial D^i\times 0\times [0, 1])$, to $G(D^i\times 0\times 0)$, and to $G(D^i\times 0\times 1)$ are tangent to $$F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$ respectively. Then $F\cup G$ extends to a concordance between $f_0$ and $f_1$. {{endthm}}

'''Proof of the Theorem \ref{thm::unknotting} using Lemma \ref{p:handle-decompose} and Lemma \ref{p:handle-add2}.(a).''' Denote by $f_0, f_1$ any two embeddings of $N_0$ into $\mathbb{R}^m$. In the following paragraph we show that there is a concordance between $f_0$ and $f_1$. From [http://www.map.mpim-bonn.mpg.de/Isotopy|Concordance Implies Isotopy Theorem] it follows that there is an isotopy between $f_0$ and $f_1$. By Lemma \ref{p:handle-decompose} there is a handle decomposition of $N_0$ with attaching maps of indices at most $n-1$. Denote by $U^l$ the manifold obtained from $\emptyset$ by the attaching the first $l$ handles, starting with $U^1\cong D^n$. Define a concordance $F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$ recursively. Take any concordance $F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$ between $f_0|_{U^1}$ and $f_1|_{U^1}$. Let us define a concordance $F_l$ between $f_0|_{U^l}$ and $f_1|_{U^l}$ using a concordance $F_{l-1}$ of $U^{l-1}$ between $f_0|_{U^{l-1}}$ and $f_1|_{U^{l-1}}$. Denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$ the $l$-th attaching map. Since $i\leq n-1$, by Lemma \ref{p:handle-add2}.(a) it follows that there is an extension of $F_{l-1}$ to a concordance $$F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$$ between the restriction of $f_0$ and $f_1$ to $U^{l-1}\cup_\phi D^i\times D^{n-i}$. {{endproof}}

'''Proof of Lemma \ref{p:handle-add2}.(a) using Lemma \ref{p:handle-add2}.(b).''' In the following text we identify $D^i\times D^{n-i}\times [0, 1]$ and $D^i\times [0, 1]\times D^{n-i}$. Define map $$\bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$$ by the formula: $$\bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right. $$ Since $$\dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$$ by general position there is an embedding $$G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$$ such that $F\bar{\phi} = G$ on $\partial (D^i \times 0\times [0, 1])$ and $F(\mbox{Int} (U\times [0, 1]))$ has a finite number of intersection points with $G(\mbox{Int}( D^i\times 0\times [0, 1]))$. Then by an isotopy $G_t$, where $G_0=G$, fixed on $\partial (D^i\times 0\times [0, 1])$ we can "push out" the self-intersection points toward $F(\partial (U\times [0, 1]))$ so that $G_1(D^i\times 0\times [0, 1])$ does not intersect $G(U\times [0, 1])$. Then we may assume that $F\cup G$ is an concordance between the restrictions of $f_0$ and $f_1$ on $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$. Denote by $A$ the $(n-i)\times (n+1)$ matrix whose rightmost $(n-i)\times (n-i)$ submatrix is the identity matrix, and whose other elements are zeroes. Denote by $v$ the field of $n-i$ normal vectors on $\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$ whose $k$-th vector has coordinates equal to the $k$-th row in $A$. Then $d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$ is the vector field tangent to $\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$. For $x\in G(\partial (D^i\times 0\times [0, 1])$ denote by $v'$ the projection of $d(F\cup f_0\cup f_1)d\phi (v(x))$ to the intersection of normal space to $G(D^i\times 0\times [0, 1])$, and tangent space to $F(\partial(U\times [0, 1]))$. Since $i == Example of non-isotopic embeddings == ; \label{sec::example} The following example is folklore. {{beginthm|Example}} \label{exm::linked_boundary} Let $N=S^k\times [0, 1]$ be the cylinder over $S^k$. (a) Then there exist non-isotopic embeddings of $N$ into $\mathbb R^{2k+1}$. (b) Then for each $a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$. (c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$. {{endthm}} {{beginproof}} ''Proof of part (b).'' Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$ be a map of degree $a$. (To prove part (a) it is sufficient to take as $h$ the identity map of $S^k$ as a map of degree one and the constant map as a map of degree zero.) Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$. Let $f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Notation_and_conventions|the standard embedding]].Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$. ''Proof of part (c).'' Clearly $l$ is well-defined. By (b) $l$ is surjective. Now take any two embeddings $f_1, f_2$ such that $l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields. Since $k\geqslant 2$ [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that there exists an isotopy of $f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$ and $f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$ and $f_2$ are isotopic. {{endproof}} [[Image:Punctured_torus_embeddings.svg|thumb|450px|(a): Embeddings $f_1$ (top) and $f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$, $i=1,2$, so the ends of the vector field define the section $s_i$; (c): embedding $s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$.]] Denote \S5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\alpha\partial$] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

Then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2]. Part (b) in the case $n>2$$n>2$ is proved in [Edwards1968, $\S$$\S$ 4, Corollary 5]. The case $n=1$$n=1$ is clear. Case $n=2$$n=2$ can be proved using the ideas presented below.

Both parts of Theorem 2.1 are special cases of Theorem 7.2.

The inequality in part (b) is sharp, see Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. Since $2(n-1) < 2n-1$$2(n-1) < 2n-1$, by general position we may assume that $g|_{X}$$g|_{X}$ is an embedding. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Alternative proof of Theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by Lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $N$$N$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial N$$\phi : \partial D^i \times D^{n-i} \to \partial N$, $f : N \to \mathbb R^m$$f : N \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(N)$$f(N)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ whose the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $i-1<2n-1 - (n-i)$$i-1<2n-1 - (n-i)$, it follows that $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$ and there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).

Lemma 3.4. Assume that $U$$U$ is a compact $n$$n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$$\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$$i\leq n-1$, $f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$$f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$ are embeddings and $F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$$F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$ is a concordance between $f_0|_U$$f_0|_U$ and $f_1|_U$$f_1|_U$.

$(a)$$(a)$ If also that $m\geq 2n$$m\geq 2n$, then there is an extension of $F$$F$ to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

$(b)$$(b)$ Assume also that there is a concordance $G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$$G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$ between $f_0|_{D^i\times 0}$$f_0|_{D^i\times 0}$ and $f_1|_{D^i\times 0}$$f_1|_{D^i\times 0}$, and on $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$ there is a field of $n-i$$n-i$ linear independed normal vectors whose restrictions to $G(\partial D^i\times 0\times [0, 1])$$G(\partial D^i\times 0\times [0, 1])$, to $G(D^i\times 0\times 0)$$G(D^i\times 0\times 0)$, and to $G(D^i\times 0\times 1)$$G(D^i\times 0\times 1)$ are tangent to

$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$ $$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$

respectively. Then $F\cup G$$F\cup G$ extends to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

Proof of the Theorem 2.2 using Lemma 3.1 and Lemma 3.4.(a). Denote by $f_0, f_1$$f_0, f_1$ any two embeddings of $N_0$$N_0$ into $\mathbb{R}^m$$\mathbb{R}^m$. In the following paragraph we show that there is a concordance between $f_0$$f_0$ and $f_1$$f_1$. From Implies Isotopy Theorem it follows that there is an isotopy between $f_0$$f_0$ and $f_1$$f_1$. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps of indices at most $n-1$$n-1$. Denote by $U^l$$U^l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching the first $l$$l$ handles, starting with $U^1\cong D^n$$U^1\cong D^n$. Define a concordance $F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$ recursively. Take any concordance $F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$ between $f_0|_{U^1}$$f_0|_{U^1}$ and $f_1|_{U^1}$$f_1|_{U^1}$. Let us define a concordance $F_l$$F_l$ between $f_0|_{U^l}$$f_0|_{U^l}$ and $f_1|_{U^l}$$f_1|_{U^l}$ using a concordance $F_{l-1}$$F_{l-1}$ of $U^{l-1}$$U^{l-1}$ between $f_0|_{U^{l-1}}$$f_0|_{U^{l-1}}$ and $f_1|_{U^{l-1}}$$f_1|_{U^{l-1}}$. Denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$$\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$ the $l$$l$-th attaching map. Since $i\leq n-1$$i\leq n-1$, by Lemma 3.4.(a) it follows that there is an extension of $F_{l-1}$$F_{l-1}$ to a concordance

$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$ $$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$$

between the restriction of $f_0$$f_0$ and $f_1$$f_1$ to $U^{l-1}\cup_\phi D^i\times D^{n-i}$$U^{l-1}\cup_\phi D^i\times D^{n-i}$.

$\square$$\square$

Proof of Lemma 3.4.(a) using Lemma 3.4.(b). In the following text we identify $D^i\times D^{n-i}\times [0, 1]$$D^i\times D^{n-i}\times [0, 1]$ and $D^i\times [0, 1]\times D^{n-i}$$D^i\times [0, 1]\times D^{n-i}$. Define map

$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$ $$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$$

by the formula:

Tex syntax error

$$\displaystyle \bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \\ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right.$$

Since

$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$ $$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$$

by general position there is an embedding

$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$ $$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$$

such that $F\bar{\phi} = G$$F\bar{\phi} = G$ on $\partial (D^i \times 0\times [0, 1])$$\partial (D^i \times 0\times [0, 1])$ and $F(\mbox{Int} (U\times [0, 1]))$$F(\mbox{Int} (U\times [0, 1]))$ has a finite number of intersection points with $G(\mbox{Int}( D^i\times 0\times [0, 1]))$$G(\mbox{Int}( D^i\times 0\times [0, 1]))$. Then by an isotopy $G_t$$G_t$, where $G_0=G$$G_0=G$, fixed on $\partial (D^i\times 0\times [0, 1])$$\partial (D^i\times 0\times [0, 1])$ we can "push out" the self-intersection points toward $F(\partial (U\times [0, 1]))$$F(\partial (U\times [0, 1]))$ so that $G_1(D^i\times 0\times [0, 1])$$G_1(D^i\times 0\times [0, 1])$ does not intersect $G(U\times [0, 1])$$G(U\times [0, 1])$. Then we may assume that $F\cup G$$F\cup G$ is an concordance between the restrictions of $f_0$$f_0$ and $f_1$$f_1$ on $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$$U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$.

Denote by $A$$A$ the $(n-i)\times (n+1)$$(n-i)\times (n+1)$ matrix whose rightmost $(n-i)\times (n-i)$$(n-i)\times (n-i)$ submatrix is the identity matrix, and whose other elements are zeroes. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$$\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$ whose $k$$k$-th vector has coordinates equal to the $k$$k$-th row in $A$$A$. Then $d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$$d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$ is the vector field tangent to $\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$$\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$. For $x\in G(\partial (D^i\times 0\times [0, 1])$$x\in G(\partial (D^i\times 0\times [0, 1])$ denote by $v'$$v'$ the projection of $d(F\cup f_0\cup f_1)d\phi (v(x))$$d(F\cup f_0\cup f_1)d\phi (v(x))$ to the intersection of normal space to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$, and tangent space to $F(\partial(U\times [0, 1]))$$F(\partial(U\times [0, 1]))$. Since $i<m+1-(n-i)$$i<m+1-(n-i)$, it follows that $\pi_{i}(V_{m+1, n-i})=0$$\pi_{i}(V_{m+1, n-i})=0$. Hence there is an extension of $v'$$v'$ to a linear independent field of vectors normal to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$. Then by Lemma 3.4.(b) there is an extension of $F\cup G$$F\cup G$ to a concordance $(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$$(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$.

(a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. (To prove part (a) it is sufficient to take as $h$$h$ the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero.) Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. There exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

Definition 5.2. Denote

$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$ $$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$ $$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

Proof of Lemma 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 5.3 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$$\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$$2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$$f(N_0)$ are linearly dependent.

Lemma 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 5.4.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$ $$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 6.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 6.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 6.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$ $$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 6.4. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$ $$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$$

is one-to-one.

Lemma 6.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$ $$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

10 References

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_k:=(1,0,\ldots,0)\in S^k$. {{beginthm|Example}} Let $N=S^k\times S^1$. Assume $k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$. {{endthm}} The surjectivity of $l$ is given analogously to Proposition \ref{exm::linked_boundary}(b). The injectivity of $l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$ and a cylinder. This example shows that Theorem \ref{thm::k_connect_classif} fails for $k=0$. {{beginthm|Example}} Let $N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$. {{endthm}} To prove the surjectivity of $l$ it is sufficient to take linked $k$-spheres in $\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres. {{beginthm|Example}} (a) Let $N_0$ be the punctured 2-torus containing the meridian $x$ and the parallel $y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$ denote by $s$ the normal field of $\epsilon$-length vectors to $f(N_0)$ defined by orientation on $N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$. (b) Let $f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$ and $l(f_2)=0$ which proves the intuitive fact that $f_1$ and $f_2$ are not isotopic. (Notice that the restrictions of $f_1$ and $f_2$ on $x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$, the values of $l(f_1)$ and $l(f_2)$ will change but will still be different (see figure (d)). {{endthm}} == Seifert linking form == ; \label{sec::linking_form} In this section assume that * $N$ is any closed orientable connected $n$-manifold, * $f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding, * if the (co)homology coefficients are omitted, then they are $\mathbb Z$, * $n$ is even and $H_1(N)$ is torsion free (these two assumptions are not required in Lemma \ref{lmm::saeki}). By $N_0$ we denote the closure of the complement in $N$ to an closed $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. The following folklore result holds. {{beginthm|Lemma}} There exists a nowhere vanishing normal vector field to $f(N_0)$. {{endthm}} {{beginproof}} There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$. A normal space to $f(N_0)$ at any point of $f(N_0)$ has dimension $n-1$. As $n$ is even thus $n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$. Since $H_1(N)$ is torsion free, it follows that $\bar e=0$. Since $N_0$ has non-empty boundary, we have that $N_0$ is homotopy equivalent to an $(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$ at any point of $f(N_0)$. Since $\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$. {{endproof}} Denote by $x, y$ two disjoint $(n-1)$-cycles in $N_0$ with integer coefficients. {{beginthm|Definition}} Denote $$L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$ where $s$ is a nowhere vanishing normal field to $f(N_0)$ and $s(x), s(y)$ are the results of the shift of $f(x), f(y)$ by $s$. {{endthm}} {{beginthm|Lemma|($L$ is well-defined)}}\label{lmm:L_well_def} The integer $L(f)(x, y)$: * is well-defined, i.e. does not change when $s$ is replaced by $s'$, * does not change when $x$ or $y$ are changed to homologous cycles and, * does not change when $f$ is changed to an isotopic embedding. {{endthm}} The first bullet was stated and proved in unpublished update of \cite{Tonkonog2010}, other two bullets are simple. {{beginthm|Lemma}}\label{lmm::saeki} Let $s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$. Then $$\mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$ where $s(y)$ is the result of the shift of $f(y)$ by $s$, and $d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$ being homotopic in the class of the nowhere vanishing vector fields. {{endthm}} This Lemma is proved in \cite[Lemma 2.2]{Saeki1999} for $n=3$, but the proof is valid in all dimensions.

'''Proof of Lemma \ref{lmm:L_well_def}.''' The first bullet point follows because: $$ \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned} $$ Here the second equality follows from Lemma \ref{lmm::saeki}. For each two homologous $(n-1)$-cycles $x, x'$ in $N_0$, the image of the homology between $x$ and $x'$ is a $n$-chain $X$ of $f(N_0)$ such that $\partial X = f(x) - f(x')$. Since $s$ is a nowhere vanishing normal field to $f(N_0)$, this implies that the supports of $s(y)$ and $X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$. Since isotopy of $f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy. {{endproof}} Lemma \ref{lmm:L_well_def} implies that $L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter. Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $. Define the dual to [[Stiefel-Whitney_characteristic_classes|Stiefel-Whitney class]] $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$ are linearly dependent. {{beginthm|Lemma}} \label{lmm::L_equality} For every $X, Y \in H_{n-1}(N_0)$ the following equality holds: $$\rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$ {{endthm}} This Lemma was stated in a unpublished update of \cite{Tonkonog2010}, the following proof is presented in \cite{Fedorov2021} using the idea from that update. See also an analogous lemma for closed manifolds in \cite[Lemma 2.2]{Crowley&Skopenkov2016}.

'''Proof of Lemma \ref{lmm::L_equality}.''' Let $-s$ be the normal field to $f(N_0)$ opposite to $s$. We get $$ \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \ &= d(s, -s)\cap X\cap Y . \end{aligned} $$ The first congruence is clear. The second equality holds because if we shift the link $s(X)\sqcup f(Y)$ by $-s$, we get the link $f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift. The third equality follows from Lemma \ref{lmm::saeki}. Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$ a general perturbation of $s$. We get: $$ \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0). $$ The first equality holds because $s'$ and $s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$ and $-s$ degenerates only at those points $x$ where $s'(x)=s(x)$. These points $x$ are exactly points where $s'(x)$ and $s(x)$ are linearly dependent. All those point $x$ form a $-cycle modulo two in $N_0$. The homotopy class of this $-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class. {{endproof}} == Classification theorems == ; \label{sec::classification} Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary. Let $N$ be a closed orientable connected $n$-manifold. By $N_0$ we denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. For a free Abelian group $A$, let $B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$ is even for each $x$ (the second condition automatically holds for n odd). {{beginthm|Definition}} For each even $n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$, where $W$ is [[Embeddings_just_below_the_stable_range:_classification#The_Whitney_invariant|Whitney invariant]], \cite[$\S]{Skopenkov2016e}. {{endthm}} {{beginthm|Lemma}} The invariant $W\Lambda$ is well-defined for $n\ge4$. {{endthm}} {{beginproof}} Note that [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that $\partial N_0$ unknots in $\mathbb R^{2n}$. Thus $f|_{\partial N_0}$ can be extended to embedding of an $n$-ball $B^n$ into $\mathbb R^{2n}$. [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting_theorems|Unknotting Spheres Theorem]] implies that $n$-sphere unknots in $\mathbb R^{2n}$. Thus all extensions of $f$ are isotopic in PL category. Note also that if $f$ and $g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$ is invariant for PL embeddings. {{endproof}} {{beginthm|Definition|of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$ is even and $H_1(N)$ is torsion-free}}\label{DefG} Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$. For each $f$ such that $W\Lambda(f)=z$ define $$ G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right) $$ where $x,y\in H_{n-1}(N_0)$. {{endthm}} Note also that $G$ depends on choice of collection $\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$. {{beginthm|Theorem}}\label{Tlink} Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. The map $$ G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}), $$ is one-to-one. {{endthm}} {{beginthm|Lemma}} For each even $n\in H_{n-1}(N)$ and each $x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$. {{endthm}} An equivalemt statement of Theorem \ref{Tlink}: {{beginthm|Theorem}}\label{thm::punctured_class} Let $N$ be a closed connected orientable $n$-manifold with $H_1(N)$ torsion-free, $n\ge 4$, $n$ even. Then (a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection. (b) The image of $L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$ is the normal Stiefel-Whitney class. {{endthm}} This is the main Theorem of \cite{Tonkonog2010} == A generalization to highly-connected manifolds == ; \label{sec::generalisations} For simplicity in this paragraph we consider only punctured manifolds, see $\S$\ref{prt::arbitrary_boundary} for a generalization. Denote by $N$ a closed $n$-manifold. By $N_0$ denote the complement in $N$ to an open $n$-ball. Thus $\partial N_0$ is the $(n-1)$-sphere. {{beginthm|Theorem}}\label{thm::k_connect_embeds} Assume that $N$ is a closed $k$-connected $n$-manifold. (a) If $n\geq 2k+3$, then $N$ embeds into $\mathbb R^{2n-k}$. (b) If $n\geq 2k+2$ and $k\geq0$, then $N_0$ embeds into $\mathbb R^{2n-k-1}$. {{endthm}} Part (a) is proved in \cite[Existence Theorem (a)]{Haefliger1961} for the Diff case and in \cite[Theorem 1.1]{Penrose&Whitehead&Zeeman1961}, \cite[Corollary 1.3]{Irwin1965} for PL case. Part (b) is proved in \cite[Corollary 4.2]{Hirsch1961a} for the Diff case and in \cite[Theorem 1.2]{Penrose&Whitehead&Zeeman1961} for the PL case. {{beginthm|Theorem}}\label{thm::k_connect_unknot} Assume that $N$ is a closed $k$-connected $n$-manifold. (a) If $m \ge 2n - k + 1$ and $n\ge2k + 2$, then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. (b) If $m \ge 2n - k$ and $n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$ into $\mathbb R^m$ are isotopic. {{endthm}} Part (a) is [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification#Unknotting theorems|Theorem 2.4]] of the survey \cite[$\S$ 2]{Skopenkov2016c}, and is proved in \cite[Corollary 2 of Theorem 24 in Chapter 8]{Zeeman1963} and \cite[Existence Theorem (b) in p. 47]{Haefliger1961}. Part (b) is proved in \cite[Theorem 10.3]{Hudson1969} for the PL case, using [[Isotopy#Concordance|concordance implies isotopy theorem]]. For $k>1$ part (b) is a corollary of Theorem \ref{thm::k_connect_classif} below. For $k=0$ part (b) coincides with Theorem \ref{thm::unknotting}b.

'''Proof of Theorem \ref{thm::k_connect_unknot}(b) for $k=1$.''' By Theorem \ref{thm::isotop_unknot} below every two immersions of $N_0$ into $\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$ for each $x\in N_0$. It follows from Theorem \ref{thm::k_connect_is_spine} that $N_0$ collapses to an $(n-2)$-dimensional subcomplex $X\subset N_0$ of some triangulation of $N_0$. By general position we may assume that $F|_{X\times[0,1]}$ is an embedding, because (n-1) < 2n$. Since $F$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$. It is clear that $f$ is isotopic to $f\circ h$ and $g$ is isotopic to $g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$ is a concordance of $f\circ h$ and $g\circ h$. By [[Isotopy#Concordance|concordance implies isotopy Theorem]] $f$ and $g$ are isotopic.{{endproof}} {{beginthm|Conjecture}} Assume that $N$ is a closed \S5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\alpha\partial$] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

Then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2]. Part (b) in the case $n>2$$n>2$ is proved in [Edwards1968, $\S$$\S$ 4, Corollary 5]. The case $n=1$$n=1$ is clear. Case $n=2$$n=2$ can be proved using the ideas presented below.

Both parts of Theorem 2.1 are special cases of Theorem 7.2.

The inequality in part (b) is sharp, see Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. Since $2(n-1) < 2n-1$$2(n-1) < 2n-1$, by general position we may assume that $g|_{X}$$g|_{X}$ is an embedding. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Alternative proof of Theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by Lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $N$$N$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial N$$\phi : \partial D^i \times D^{n-i} \to \partial N$, $f : N \to \mathbb R^m$$f : N \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(N)$$f(N)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ whose the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $i-1<2n-1 - (n-i)$$i-1<2n-1 - (n-i)$, it follows that $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$ and there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).

Lemma 3.4. Assume that $U$$U$ is a compact $n$$n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$$\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$$i\leq n-1$, $f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$$f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$ are embeddings and $F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$$F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$ is a concordance between $f_0|_U$$f_0|_U$ and $f_1|_U$$f_1|_U$.

$(a)$$(a)$ If also that $m\geq 2n$$m\geq 2n$, then there is an extension of $F$$F$ to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

$(b)$$(b)$ Assume also that there is a concordance $G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$$G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$ between $f_0|_{D^i\times 0}$$f_0|_{D^i\times 0}$ and $f_1|_{D^i\times 0}$$f_1|_{D^i\times 0}$, and on $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$ there is a field of $n-i$$n-i$ linear independed normal vectors whose restrictions to $G(\partial D^i\times 0\times [0, 1])$$G(\partial D^i\times 0\times [0, 1])$, to $G(D^i\times 0\times 0)$$G(D^i\times 0\times 0)$, and to $G(D^i\times 0\times 1)$$G(D^i\times 0\times 1)$ are tangent to

$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$ $$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$

respectively. Then $F\cup G$$F\cup G$ extends to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

Proof of the Theorem 2.2 using Lemma 3.1 and Lemma 3.4.(a). Denote by $f_0, f_1$$f_0, f_1$ any two embeddings of $N_0$$N_0$ into $\mathbb{R}^m$$\mathbb{R}^m$. In the following paragraph we show that there is a concordance between $f_0$$f_0$ and $f_1$$f_1$. From Implies Isotopy Theorem it follows that there is an isotopy between $f_0$$f_0$ and $f_1$$f_1$. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps of indices at most $n-1$$n-1$. Denote by $U^l$$U^l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching the first $l$$l$ handles, starting with $U^1\cong D^n$$U^1\cong D^n$. Define a concordance $F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$ recursively. Take any concordance $F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$ between $f_0|_{U^1}$$f_0|_{U^1}$ and $f_1|_{U^1}$$f_1|_{U^1}$. Let us define a concordance $F_l$$F_l$ between $f_0|_{U^l}$$f_0|_{U^l}$ and $f_1|_{U^l}$$f_1|_{U^l}$ using a concordance $F_{l-1}$$F_{l-1}$ of $U^{l-1}$$U^{l-1}$ between $f_0|_{U^{l-1}}$$f_0|_{U^{l-1}}$ and $f_1|_{U^{l-1}}$$f_1|_{U^{l-1}}$. Denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$$\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$ the $l$$l$-th attaching map. Since $i\leq n-1$$i\leq n-1$, by Lemma 3.4.(a) it follows that there is an extension of $F_{l-1}$$F_{l-1}$ to a concordance

$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$ $$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$$

between the restriction of $f_0$$f_0$ and $f_1$$f_1$ to $U^{l-1}\cup_\phi D^i\times D^{n-i}$$U^{l-1}\cup_\phi D^i\times D^{n-i}$.

$\square$$\square$

Proof of Lemma 3.4.(a) using Lemma 3.4.(b). In the following text we identify $D^i\times D^{n-i}\times [0, 1]$$D^i\times D^{n-i}\times [0, 1]$ and $D^i\times [0, 1]\times D^{n-i}$$D^i\times [0, 1]\times D^{n-i}$. Define map

$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$ $$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$$

by the formula:

Tex syntax error

$$\displaystyle \bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \\ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right.$$

Since

$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$ $$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$$

by general position there is an embedding

$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$ $$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$$

such that $F\bar{\phi} = G$$F\bar{\phi} = G$ on $\partial (D^i \times 0\times [0, 1])$$\partial (D^i \times 0\times [0, 1])$ and $F(\mbox{Int} (U\times [0, 1]))$$F(\mbox{Int} (U\times [0, 1]))$ has a finite number of intersection points with $G(\mbox{Int}( D^i\times 0\times [0, 1]))$$G(\mbox{Int}( D^i\times 0\times [0, 1]))$. Then by an isotopy $G_t$$G_t$, where $G_0=G$$G_0=G$, fixed on $\partial (D^i\times 0\times [0, 1])$$\partial (D^i\times 0\times [0, 1])$ we can "push out" the self-intersection points toward $F(\partial (U\times [0, 1]))$$F(\partial (U\times [0, 1]))$ so that $G_1(D^i\times 0\times [0, 1])$$G_1(D^i\times 0\times [0, 1])$ does not intersect $G(U\times [0, 1])$$G(U\times [0, 1])$. Then we may assume that $F\cup G$$F\cup G$ is an concordance between the restrictions of $f_0$$f_0$ and $f_1$$f_1$ on $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$$U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$.

Denote by $A$$A$ the $(n-i)\times (n+1)$$(n-i)\times (n+1)$ matrix whose rightmost $(n-i)\times (n-i)$$(n-i)\times (n-i)$ submatrix is the identity matrix, and whose other elements are zeroes. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$$\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$ whose $k$$k$-th vector has coordinates equal to the $k$$k$-th row in $A$$A$. Then $d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$$d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$ is the vector field tangent to $\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$$\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$. For $x\in G(\partial (D^i\times 0\times [0, 1])$$x\in G(\partial (D^i\times 0\times [0, 1])$ denote by $v'$$v'$ the projection of $d(F\cup f_0\cup f_1)d\phi (v(x))$$d(F\cup f_0\cup f_1)d\phi (v(x))$ to the intersection of normal space to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$, and tangent space to $F(\partial(U\times [0, 1]))$$F(\partial(U\times [0, 1]))$. Since $i<m+1-(n-i)$$i<m+1-(n-i)$, it follows that $\pi_{i}(V_{m+1, n-i})=0$$\pi_{i}(V_{m+1, n-i})=0$. Hence there is an extension of $v'$$v'$ to a linear independent field of vectors normal to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$. Then by Lemma 3.4.(b) there is an extension of $F\cup G$$F\cup G$ to a concordance $(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$$(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$.

(a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. (To prove part (a) it is sufficient to take as $h$$h$ the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero.) Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. There exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

Definition 5.2. Denote

$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$ $$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$ $$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

Proof of Lemma 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 5.3 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$$\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$$2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$$f(N_0)$ are linearly dependent.

Lemma 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 5.4.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$ $$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 6.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 6.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 6.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$ $$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 6.4. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$ $$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$$

is one-to-one.

Lemma 6.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$ $$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

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$-connected $-manifold. Then any two embeddings of $N_0$ in $\mathbb R^7$ are isotopic. {{endthm}} We may hope to get around the restrictions of Theorem \ref{thm::k_connect_is_spine} using the [[Some_calculations_involving_configuration_spaces_of_distinct_points|deleted product criterion]]. {{beginthm|Theorem}}\label{thm::k_connect_classif} Assume $N$ is a closed $k$-connected $n$-manifold. Then for each $k\ge1$ there exists a bijection $$ W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}), $$ where $\mathbb Z_{(s)}$ denote $\mathbb Z$ for $s$ even and $\mathbb Z_2$ for $s$ odd. {{endthm}} For definition of $W_0'$ and the proof of the latter Theorem see \cite[Lemma 2.2($W_0'$)]{Skopenkov2010}. The latter theorem was essentially proven in \cite[Theorem 2.1]{Vrabec1989}. Latter Theorem is essentially known result. Compare to the Theorem \ref{thm::punctured_class}, which describes $\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case. == Comments on non-spherical boundary == ; \label{prt::arbitrary_boundary} {{beginthm|Theorem}}\label{thm::arbitarty_k_connect_embeds} Assume that $N$ is a compact $k$-connected $n$-manifold, $\partial N\neq\emptyset$, $(N, \partial N)$ is $k$-connected and $k< n - 3$. Then $N$ embeds into $\mathbb R^{2n-k-1}$. {{endthm}} This is \cite[Theorem on p.567]{Wall1965}. {{beginproof}} By Theorem \ref{thm::k_immersible} below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$. Since $N$ is $k$-connected it follows from Theorem \ref{thm::k_connect_is_spine} that $N$ collapses to an $(n-k-1)$-dimensional subcomplex $X\subset N$ of some triangulation of $N$. By general position we may assume that $f|_X$ is an embedding, because (n-k) < 2n-k-1$. Since $f$ is an immersion, it follows that $X$ has a sufficiently small regular neighbourhood $M\supset X$ such that $f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$. It is clear that $f\circ h$ is an embedding.{{endproof}} {{beginthm|Theorem}}\label{thm::arbitrary_k_connect_unknotting} Assume that $N$ is a $n$-manifold. If $N$ has $(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$, $m \ge 2n - k$, then any two embeddings of $N$ into $\mathbb R^m$ are isotopic. {{endthm}} Proof is similar to the proof of theorem \ref{thm::k_connect_unknot}. For a compact connected $n$-manifold with boundary, the property of having an $(n − k − 1)$-dimensional spine is close to $k$-connectedness. Indeed, the following theorem holds. {{beginthm|Theorem}}\label{thm::k_connect_is_spine} Every compact connected $n$-manifold $N$ with boundary for which $(N, \partial N)$ is $k$-connected, $\pi_1(\partial N)=0$, $k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$-dimensional spine. {{endthm}} For this result see \cite[Theorem 5.5]{Wall1964a} and \cite[Lemma 5.1 and Remark 5.2]{Horvatic1969}. See also valuable remarks in \cite{Levine&Lidman2018} and \cite{Skopenkov2019}. == Comments on immersions == ; {{beginthm|Theorem}}[Smale-Hirsch; \cite{Hirsch1959} and \cite{Haefliger&Poenaru1964}] The space of immersions of a manifold in $\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$ to $\R^m$. {{endthm}} {{beginthm|Theorem}}[\cite[Theorem 6.4]{Hirsch1959}]\label{thm::imm_dec_dim} If $N$ is immersible in $\R^{m+r}$ with a normal $r$-field, then $N$ is immersible in $\R^m$. {{endthm}} {{beginthm|Theorem}} Every $n$-manifold $N$ with non-empty boundary is immersible in $\R^{2n-1}$. {{endthm}} {{beginthm|Theorem}}[Whitney; \cite[Theorem 6.6]{Hirsch1961a}] Every $n$-manifold $N$ is immersible in $\R^{2n-1}$. {{endthm}} Denote by $V_{m,n}$ is Stiefel manifold of $n$-frames in $\R^m$. {{beginthm|Theorem}}\label{thm::k_immersible} Suppose $N$ is a $n$-manifold with non-empty boudary, $(N,\partial N)$ is $k$-connected. Then $N$ is immersible in $\R^m$ for each $m\geq2n-k-1$. {{endthm}} {{beginproof}} It suffices to show that exists an immersion of $N$ in $\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$ to $\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$. It is clear that a linear monomorphism exists on $\S$5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\alpha\partial$] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

Then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2]. Part (b) in the case $n>2$$n>2$ is proved in [Edwards1968, $\S$$\S$ 4, Corollary 5]. The case $n=1$$n=1$ is clear. Case $n=2$$n=2$ can be proved using the ideas presented below.

Both parts of Theorem 2.1 are special cases of Theorem 7.2.

The inequality in part (b) is sharp, see Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. Since $2(n-1) < 2n-1$$2(n-1) < 2n-1$, by general position we may assume that $g|_{X}$$g|_{X}$ is an embedding. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Alternative proof of Theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by Lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $N$$N$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial N$$\phi : \partial D^i \times D^{n-i} \to \partial N$, $f : N \to \mathbb R^m$$f : N \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(N)$$f(N)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ whose the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $i-1<2n-1 - (n-i)$$i-1<2n-1 - (n-i)$, it follows that $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$ and there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).

Lemma 3.4. Assume that $U$$U$ is a compact $n$$n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$$\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$$i\leq n-1$, $f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$$f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$ are embeddings and $F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$$F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$ is a concordance between $f_0|_U$$f_0|_U$ and $f_1|_U$$f_1|_U$.

$(a)$$(a)$ If also that $m\geq 2n$$m\geq 2n$, then there is an extension of $F$$F$ to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

$(b)$$(b)$ Assume also that there is a concordance $G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$$G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$ between $f_0|_{D^i\times 0}$$f_0|_{D^i\times 0}$ and $f_1|_{D^i\times 0}$$f_1|_{D^i\times 0}$, and on $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$ there is a field of $n-i$$n-i$ linear independed normal vectors whose restrictions to $G(\partial D^i\times 0\times [0, 1])$$G(\partial D^i\times 0\times [0, 1])$, to $G(D^i\times 0\times 0)$$G(D^i\times 0\times 0)$, and to $G(D^i\times 0\times 1)$$G(D^i\times 0\times 1)$ are tangent to

$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$ $$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$

respectively. Then $F\cup G$$F\cup G$ extends to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

Proof of the Theorem 2.2 using Lemma 3.1 and Lemma 3.4.(a). Denote by $f_0, f_1$$f_0, f_1$ any two embeddings of $N_0$$N_0$ into $\mathbb{R}^m$$\mathbb{R}^m$. In the following paragraph we show that there is a concordance between $f_0$$f_0$ and $f_1$$f_1$. From Implies Isotopy Theorem it follows that there is an isotopy between $f_0$$f_0$ and $f_1$$f_1$. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps of indices at most $n-1$$n-1$. Denote by $U^l$$U^l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching the first $l$$l$ handles, starting with $U^1\cong D^n$$U^1\cong D^n$. Define a concordance $F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$ recursively. Take any concordance $F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$ between $f_0|_{U^1}$$f_0|_{U^1}$ and $f_1|_{U^1}$$f_1|_{U^1}$. Let us define a concordance $F_l$$F_l$ between $f_0|_{U^l}$$f_0|_{U^l}$ and $f_1|_{U^l}$$f_1|_{U^l}$ using a concordance $F_{l-1}$$F_{l-1}$ of $U^{l-1}$$U^{l-1}$ between $f_0|_{U^{l-1}}$$f_0|_{U^{l-1}}$ and $f_1|_{U^{l-1}}$$f_1|_{U^{l-1}}$. Denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$$\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$ the $l$$l$-th attaching map. Since $i\leq n-1$$i\leq n-1$, by Lemma 3.4.(a) it follows that there is an extension of $F_{l-1}$$F_{l-1}$ to a concordance

$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$ $$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$$

between the restriction of $f_0$$f_0$ and $f_1$$f_1$ to $U^{l-1}\cup_\phi D^i\times D^{n-i}$$U^{l-1}\cup_\phi D^i\times D^{n-i}$.

$\square$$\square$

Proof of Lemma 3.4.(a) using Lemma 3.4.(b). In the following text we identify $D^i\times D^{n-i}\times [0, 1]$$D^i\times D^{n-i}\times [0, 1]$ and $D^i\times [0, 1]\times D^{n-i}$$D^i\times [0, 1]\times D^{n-i}$. Define map

$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$ $$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$$

by the formula:

Tex syntax error

$$\displaystyle \bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \\ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right.$$

Since

$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$ $$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$$

by general position there is an embedding

$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$ $$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$$

such that $F\bar{\phi} = G$$F\bar{\phi} = G$ on $\partial (D^i \times 0\times [0, 1])$$\partial (D^i \times 0\times [0, 1])$ and $F(\mbox{Int} (U\times [0, 1]))$$F(\mbox{Int} (U\times [0, 1]))$ has a finite number of intersection points with $G(\mbox{Int}( D^i\times 0\times [0, 1]))$$G(\mbox{Int}( D^i\times 0\times [0, 1]))$. Then by an isotopy $G_t$$G_t$, where $G_0=G$$G_0=G$, fixed on $\partial (D^i\times 0\times [0, 1])$$\partial (D^i\times 0\times [0, 1])$ we can "push out" the self-intersection points toward $F(\partial (U\times [0, 1]))$$F(\partial (U\times [0, 1]))$ so that $G_1(D^i\times 0\times [0, 1])$$G_1(D^i\times 0\times [0, 1])$ does not intersect $G(U\times [0, 1])$$G(U\times [0, 1])$. Then we may assume that $F\cup G$$F\cup G$ is an concordance between the restrictions of $f_0$$f_0$ and $f_1$$f_1$ on $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$$U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$.

Denote by $A$$A$ the $(n-i)\times (n+1)$$(n-i)\times (n+1)$ matrix whose rightmost $(n-i)\times (n-i)$$(n-i)\times (n-i)$ submatrix is the identity matrix, and whose other elements are zeroes. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$$\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$ whose $k$$k$-th vector has coordinates equal to the $k$$k$-th row in $A$$A$. Then $d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$$d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$ is the vector field tangent to $\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$$\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$. For $x\in G(\partial (D^i\times 0\times [0, 1])$$x\in G(\partial (D^i\times 0\times [0, 1])$ denote by $v'$$v'$ the projection of $d(F\cup f_0\cup f_1)d\phi (v(x))$$d(F\cup f_0\cup f_1)d\phi (v(x))$ to the intersection of normal space to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$, and tangent space to $F(\partial(U\times [0, 1]))$$F(\partial(U\times [0, 1]))$. Since $i<m+1-(n-i)$$i<m+1-(n-i)$, it follows that $\pi_{i}(V_{m+1, n-i})=0$$\pi_{i}(V_{m+1, n-i})=0$. Hence there is an extension of $v'$$v'$ to a linear independent field of vectors normal to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$. Then by Lemma 3.4.(b) there is an extension of $F\cup G$$F\cup G$ to a concordance $(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$$(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$.

(a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. (To prove part (a) it is sufficient to take as $h$$h$ the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero.) Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. There exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

Definition 5.2. Denote

$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$ $$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$ $$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

Proof of Lemma 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 5.3 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$$\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$$2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$$f(N_0)$ are linearly dependent.

Lemma 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 5.4.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$ $$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 6.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 6.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 6.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$ $$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 6.4. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$ $$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$$

is one-to-one.

Lemma 6.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$ $$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

10 References

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$-skeleton of $N$. The obstruction to extend the linear monomorphism from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$. For $r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$ is $k$-connected and has non-empty boundary. Thus the obstruction is always zero and such linear monomorphism exists. {{endproof}} {{beginthm|Theorem}}\label{thm::isotop_unknot} Suppose $N$ is a connected $n$-manifold with non-empty boudary, $(N, \partial N)$ is $k$-connected and $m\geq2n-k$. Then every two immersions of $N$ in $\R^m$ are regulary homotopic. {{endthm}} {{beginproof}} It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$ to $\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$-skeleton of $N$. It is clear that homotopy exists on $\S$5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\alpha\partial$] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

Then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2]. Part (b) in the case $n>2$$n>2$ is proved in [Edwards1968, $\S$$\S$ 4, Corollary 5]. The case $n=1$$n=1$ is clear. Case $n=2$$n=2$ can be proved using the ideas presented below.

Both parts of Theorem 2.1 are special cases of Theorem 7.2.

The inequality in part (b) is sharp, see Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. Since $2(n-1) < 2n-1$$2(n-1) < 2n-1$, by general position we may assume that $g|_{X}$$g|_{X}$ is an embedding. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Alternative proof of Theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by Lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $N$$N$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial N$$\phi : \partial D^i \times D^{n-i} \to \partial N$, $f : N \to \mathbb R^m$$f : N \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(N)$$f(N)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ whose the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $i-1<2n-1 - (n-i)$$i-1<2n-1 - (n-i)$, it follows that $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$ and there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).

Lemma 3.4. Assume that $U$$U$ is a compact $n$$n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$$\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$$i\leq n-1$, $f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$$f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$ are embeddings and $F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$$F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$ is a concordance between $f_0|_U$$f_0|_U$ and $f_1|_U$$f_1|_U$.

$(a)$$(a)$ If also that $m\geq 2n$$m\geq 2n$, then there is an extension of $F$$F$ to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

$(b)$$(b)$ Assume also that there is a concordance $G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$$G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$ between $f_0|_{D^i\times 0}$$f_0|_{D^i\times 0}$ and $f_1|_{D^i\times 0}$$f_1|_{D^i\times 0}$, and on $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$ there is a field of $n-i$$n-i$ linear independed normal vectors whose restrictions to $G(\partial D^i\times 0\times [0, 1])$$G(\partial D^i\times 0\times [0, 1])$, to $G(D^i\times 0\times 0)$$G(D^i\times 0\times 0)$, and to $G(D^i\times 0\times 1)$$G(D^i\times 0\times 1)$ are tangent to

$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$ $$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$

respectively. Then $F\cup G$$F\cup G$ extends to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

Proof of the Theorem 2.2 using Lemma 3.1 and Lemma 3.4.(a). Denote by $f_0, f_1$$f_0, f_1$ any two embeddings of $N_0$$N_0$ into $\mathbb{R}^m$$\mathbb{R}^m$. In the following paragraph we show that there is a concordance between $f_0$$f_0$ and $f_1$$f_1$. From Implies Isotopy Theorem it follows that there is an isotopy between $f_0$$f_0$ and $f_1$$f_1$. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps of indices at most $n-1$$n-1$. Denote by $U^l$$U^l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching the first $l$$l$ handles, starting with $U^1\cong D^n$$U^1\cong D^n$. Define a concordance $F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$ recursively. Take any concordance $F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$ between $f_0|_{U^1}$$f_0|_{U^1}$ and $f_1|_{U^1}$$f_1|_{U^1}$. Let us define a concordance $F_l$$F_l$ between $f_0|_{U^l}$$f_0|_{U^l}$ and $f_1|_{U^l}$$f_1|_{U^l}$ using a concordance $F_{l-1}$$F_{l-1}$ of $U^{l-1}$$U^{l-1}$ between $f_0|_{U^{l-1}}$$f_0|_{U^{l-1}}$ and $f_1|_{U^{l-1}}$$f_1|_{U^{l-1}}$. Denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$$\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$ the $l$$l$-th attaching map. Since $i\leq n-1$$i\leq n-1$, by Lemma 3.4.(a) it follows that there is an extension of $F_{l-1}$$F_{l-1}$ to a concordance

$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$ $$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$$

between the restriction of $f_0$$f_0$ and $f_1$$f_1$ to $U^{l-1}\cup_\phi D^i\times D^{n-i}$$U^{l-1}\cup_\phi D^i\times D^{n-i}$.

$\square$$\square$

Proof of Lemma 3.4.(a) using Lemma 3.4.(b). In the following text we identify $D^i\times D^{n-i}\times [0, 1]$$D^i\times D^{n-i}\times [0, 1]$ and $D^i\times [0, 1]\times D^{n-i}$$D^i\times [0, 1]\times D^{n-i}$. Define map

$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$ $$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$$

by the formula:

Tex syntax error

$$\displaystyle \bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \\ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right.$$

Since

$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$ $$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$$

by general position there is an embedding

$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$ $$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$$

such that $F\bar{\phi} = G$$F\bar{\phi} = G$ on $\partial (D^i \times 0\times [0, 1])$$\partial (D^i \times 0\times [0, 1])$ and $F(\mbox{Int} (U\times [0, 1]))$$F(\mbox{Int} (U\times [0, 1]))$ has a finite number of intersection points with $G(\mbox{Int}( D^i\times 0\times [0, 1]))$$G(\mbox{Int}( D^i\times 0\times [0, 1]))$. Then by an isotopy $G_t$$G_t$, where $G_0=G$$G_0=G$, fixed on $\partial (D^i\times 0\times [0, 1])$$\partial (D^i\times 0\times [0, 1])$ we can "push out" the self-intersection points toward $F(\partial (U\times [0, 1]))$$F(\partial (U\times [0, 1]))$ so that $G_1(D^i\times 0\times [0, 1])$$G_1(D^i\times 0\times [0, 1])$ does not intersect $G(U\times [0, 1])$$G(U\times [0, 1])$. Then we may assume that $F\cup G$$F\cup G$ is an concordance between the restrictions of $f_0$$f_0$ and $f_1$$f_1$ on $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$$U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$.

Denote by $A$$A$ the $(n-i)\times (n+1)$$(n-i)\times (n+1)$ matrix whose rightmost $(n-i)\times (n-i)$$(n-i)\times (n-i)$ submatrix is the identity matrix, and whose other elements are zeroes. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$$\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$ whose $k$$k$-th vector has coordinates equal to the $k$$k$-th row in $A$$A$. Then $d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$$d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$ is the vector field tangent to $\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$$\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$. For $x\in G(\partial (D^i\times 0\times [0, 1])$$x\in G(\partial (D^i\times 0\times [0, 1])$ denote by $v'$$v'$ the projection of $d(F\cup f_0\cup f_1)d\phi (v(x))$$d(F\cup f_0\cup f_1)d\phi (v(x))$ to the intersection of normal space to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$, and tangent space to $F(\partial(U\times [0, 1]))$$F(\partial(U\times [0, 1]))$. Since $i<m+1-(n-i)$$i<m+1-(n-i)$, it follows that $\pi_{i}(V_{m+1, n-i})=0$$\pi_{i}(V_{m+1, n-i})=0$. Hence there is an extension of $v'$$v'$ to a linear independent field of vectors normal to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$. Then by Lemma 3.4.(b) there is an extension of $F\cup G$$F\cup G$ to a concordance $(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$$(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$.

(a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. (To prove part (a) it is sufficient to take as $h$$h$ the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero.) Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. There exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

Definition 5.2. Denote

$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$ $$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$ $$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

Proof of Lemma 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 5.3 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$$\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$$2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$$f(N_0)$ are linearly dependent.

Lemma 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 5.4.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$ $$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 6.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 6.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 6.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$ $$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 6.4. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$ $$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$$

is one-to-one.

Lemma 6.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$ $$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

10 References

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$-skeleton of $N$. The obstruction to extend the homotopy from $(r-1)$-skeleton to $r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$. For $r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$ is $k$-connected and $N$ has non-empty boundary. Thus the obstruction is always zero and such homotopy of linear monomorphisms exists. {{endproof}} == References == {{#RefList:}} [[Category:Manifolds]] [[Category:Embeddings of manifolds]]\S5 we introduce an invariant of embedding of a $n$$n$-manifold in $(n-1)$$(n-1)$-space for even $n$$n$. In $\S$$\S$7 which is independent from $\S$$\S$4, $\S$$\S$5 and $\S$$\S$6 we state generalisations of theorems from $\S$$\S$2 to highly-connected manifolds.

For a general introduction to embeddings as well as the notation and conventions used on this page, we refer to [Skopenkov2016c, $\S$$\S$1, $\S$$\S$3]. In those pages mostly results for closed manifolds are stated.

If the category is omitted, then we assume the smooth (DIFF) category. Denote $\mathrm{Emb}^mN$$\mathrm{Emb}^mN$ the set of all embeddings $f\colon N\to\mathbb R^m$$f\colon N\to\mathbb R^m$ up to isotopy. We denote by $\mathrm{lk}$$\mathrm{lk}$ the linking coefficient [Seifert&Threlfall1980, $\S$$\S$ 77] of two disjoint cycles.

We state the simplest results. These results can be deduced from the Haefliger-Weber deleted product criterion [Skopenkov2006, $\S$$\S$ 5], see [Haefliger1963, 6.4], [Skopenkov2002, Theorem 1.1$\alpha\partial$$\alpha\partial$] for the DIFF case and [Skopenkov2002, Theorem 1.3$\alpha\partial$$\alpha\partial$] for the PL case. For some results we present direct proofs, which are easier than deduction from this criterion.

We do not claim the references we give are references to original proofs.

2 Embedding and unknotting theorems

Theorem 2.1. Assume that $N$$N$ is a compact connected $n$$n$-manifold.

(a) Then $N$$N$ embeds into $\mathbb R^{2n}$$\mathbb R^{2n}$.

(b) If $N$$N$ has non-empty boundary, then $N$$N$ embeds into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$.

Part (a) is well-known strong Whitney embedding theorem.

Theorem 2.2. Assume that $N$$N$ is a compact connected $n$$n$-manifold and either

(a) $m \ge 2n+1 \ge 5$$m \ge 2n+1 \ge 5$ or

(b) $N$$N$ has non-empty boundary and $m\geq 2n$$m\geq 2n$.

Then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Whitney-Wu Unknotting Theorem, see [Skopenkov2016c, $\S$$\S$ 2, Theorems 2.1, 2.2]. Part (b) in the case $n>2$$n>2$ is proved in [Edwards1968, $\S$$\S$ 4, Corollary 5]. The case $n=1$$n=1$ is clear. Case $n=2$$n=2$ can be proved using the ideas presented below.

Both parts of Theorem 2.1 are special cases of Theorem 7.2.

The inequality in part (b) is sharp, see Proposition 4.1.

These basic results can be generalized to highly-connected manifolds (see $\S$$\S$7). All stated theorems of $\S$$\S$2 and $\S$$\S$7 for manifolds with non-empty boundary can be proved using analogous results for immersions of manifolds and general position ideas.

3 Proofs of Theorem 2.1.b and Theorem 2.2.b

The first proof of Theorem 2.1.b uses immersions, while the second does not.

First proof of Theorem 2.1.b. By the strong Whitney immersion theorem there exist an immersion $g\colon N\to\mathbb R^{2n-1}$$g\colon N\to\mathbb R^{2n-1}$. Since $N$$N$ is connected and has non-empty boundary, it follows that $N$$N$ collapses to an $(n-1)$$(n-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. Since $2(n-1) < 2n-1$$2(n-1) < 2n-1$, by general position we may assume that $g|_{X}$$g|_{X}$ is an embedding. Since $g$$g$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $g|_{M}$$g|_{M}$ is embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. The composition $g\circ h$$g\circ h$ is an embedding of $N$$N$.

$\square$$\square$

This proof is essentially contained in [Hirsch1961a, Theorem 4.6] for the Diff case and in references for Theorem 7.1 below or in [Horvatic1971, Theorem 5.2] for the PL case.

For the alternative proof we need some lemmas.

Lemma 3.1. [Wall1966] Assume that $N$$N$ is a closed connected smooth $n$$n$-manifold. Then $N_0$$N_0$ have handle decomposition with indices of attaching map at most $n-1$$n-1$.

Lemma 3.2. Assume that $N$$N$ is a closed smooth $n$$n$-manifold and $\phi:\partial D^i\times D^{n-i}\to \partial N$$\phi:\partial D^i\times D^{n-i}\to \partial N$ is an attaching map such that $i<n$$i<n$. If there is embedding $f:N \to \mathbb R^{2n-1}$$f:N \to \mathbb R^{2n-1}$, then $f$$f$ extends to an embedding of $N\cup_{\phi} D^i\times D^{n-i}$$N\cup_{\phi} D^i\times D^{n-i}$.

Alternative proof of Theorem 2.1.b. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps $\phi_1,\ldots,\phi_s$$\phi_1,\ldots,\phi_s$ of indices smaller than $n$$n$. Denote by $U_l$$U_l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching first $l$$l$ handles. Take any embedding $F_1:U_1 \cong D^n\to \R^{2n-1}$$F_1:U_1 \cong D^n\to \R^{2n-1}$. Let us define an embedding $F_l$$F_l$ of $U_l$$U_l$ using an embedding $F_{l-1}$$F_{l-1}$ of $U_{l-1}$$U_{l-1}$. Since the index $i$$i$ of $\phi_l$$\phi_l$ is smaller than $n$$n$, by Lemma 3.2 there is extension of $F_{l-1}$$F_{l-1}$ to an embedding $F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$$F_l:U_{l-1}\cup_{\phi_l} D^i\times D^{n-i}\to\R^{2n-1}$.

$\square$$\square$

Lemma 3.3. Let $N$$N$ be a closed smooth $n$$n$-manifold and $\phi : \partial D^i \times D^{n-i} \to \partial N$$\phi : \partial D^i \times D^{n-i} \to \partial N$, $f : N \to \mathbb R^m$$f : N \to \mathbb R^m$, $g : D^i\times 0 \to \mathbb R^m$$g : D^i\times 0 \to \mathbb R^m$, are smooth embeddings such that $f\phi = g$$f\phi = g$ on $\partial D^i \times 0$$\partial D^i \times 0$. Suppose that on $f(D^i\times 0)$$f(D^i\times 0)$ there is a field of $n-i$$n-i$ pairwise orthogonal normal vectors whose restriction to $f(\partial D^i\times 0)$$f(\partial D^i\times 0)$ is tangent to $f(N)$$f(N)$. Then $f\cup g$$f\cup g$ extends to a smooth embedding $M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$$M \cup_{\phi_i} D^i\times D^{n-i} \to\ \mathbb R^m$.

Proof of Lemma 3.2 using Lemma 3.3. Since $2i+1\leq 2n-1$$2i+1\leq 2n-1$, it follows that there is an extension of $f$$f$ to embedding $g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$$g:N\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0\to \mathbb R^{2n-1}$. Denote by $V$$V$ the $(n-i)\times n$$(n-i)\times n$ matrix such that the right most $(n-i)\times (n-i)$$(n-i)\times (n-i)$ block is the identity matrix and other elements equal to zero. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial D^i\times 0\subset D^i\times D^{n-i}$$\partial D^i\times 0\subset D^i\times D^{n-i}$ whose the $k$$k$-th vector have coordinates equal to the $k$$k$-th row in $V$$V$. Then $d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$$d\phi (v)= (d\phi (v_1),\ldots , d\phi (v_{n-i}))$ is vector field tangent to $\partial N$$\partial N$. Denote by $v'$$v'$ the projection of $(df\circ d\phi)|_{\partial D^i\times 0}(v)$$(df\circ d\phi)|_{\partial D^i\times 0}(v)$ to the intersection of normal space to $g(\partial D^i\times 0)$$g(\partial D^i\times 0)$ and tangent space to $\partial N$$\partial N$. Since $i-1<2n-1 - (n-i)$$i-1<2n-1 - (n-i)$, it follows that $\pi_{i-1}(V_{2n-1, n-i})=0$$\pi_{i-1}(V_{2n-1, n-i})=0$ and there is an extension of $v'$$v'$ to an orthonormal field of vectors normal to $g(D^i\times 0)$$g(D^i\times 0)$. Then by Lemma 3.3 there is an extension of $f\cup g|_{D^i\times 0}$$f\cup g|_{D^i\times 0}$ to $N \cup_{\phi_i} D^i\times D^{n-i}$$N \cup_{\phi_i} D^i\times D^{n-i}$.

$\square$$\square$

In the proof of Theorem 2.2 we will use Lemma 3.1 and Lemma 3.4.(a).

Lemma 3.4. Assume that $U$$U$ is a compact $n$$n$-manifold, $\phi:\partial D^i\times D^{n-i}\to \partial U$$\phi:\partial D^i\times D^{n-i}\to \partial U$ is an embedding with $i\leq n-1$$i\leq n-1$, $f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$$f_0, f_1: U\cup_{\phi} D^i\times D^{n-i}\to \mathbb{R}^m$ are embeddings and $F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$$F:U\times [0, 1]\to \mathbb{R}^m\times [0, 1]$ is a concordance between $f_0|_U$$f_0|_U$ and $f_1|_U$$f_1|_U$.

$(a)$$(a)$ If also that $m\geq 2n$$m\geq 2n$, then there is an extension of $F$$F$ to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

$(b)$$(b)$ Assume also that there is a concordance $G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$$G:D^i\times 0\times [0, 1]:\mathbb{R}^m\times [0, 1]$ between $f_0|_{D^i\times 0}$$f_0|_{D^i\times 0}$ and $f_1|_{D^i\times 0}$$f_1|_{D^i\times 0}$, and on $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$ there is a field of $n-i$$n-i$ linear independed normal vectors whose restrictions to $G(\partial D^i\times 0\times [0, 1])$$G(\partial D^i\times 0\times [0, 1])$, to $G(D^i\times 0\times 0)$$G(D^i\times 0\times 0)$, and to $G(D^i\times 0\times 1)$$G(D^i\times 0\times 1)$ are tangent to

$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$ $$\displaystyle F(U\times [0, 1])\quad\text{to}\quad f_0(D^i\times D^{n-i})\times 0,\quad\text{and to}\quad f_1(D^i\times D^{n-i})\times 1,$$

respectively. Then $F\cup G$$F\cup G$ extends to a concordance between $f_0$$f_0$ and $f_1$$f_1$.

Proof of the Theorem 2.2 using Lemma 3.1 and Lemma 3.4.(a). Denote by $f_0, f_1$$f_0, f_1$ any two embeddings of $N_0$$N_0$ into $\mathbb{R}^m$$\mathbb{R}^m$. In the following paragraph we show that there is a concordance between $f_0$$f_0$ and $f_1$$f_1$. From Implies Isotopy Theorem it follows that there is an isotopy between $f_0$$f_0$ and $f_1$$f_1$. By Lemma 3.1 there is a handle decomposition of $N_0$$N_0$ with attaching maps of indices at most $n-1$$n-1$. Denote by $U^l$$U^l$ the manifold obtained from $\emptyset$$\emptyset$ by the attaching the first $l$$l$ handles, starting with $U^1\cong D^n$$U^1\cong D^n$. Define a concordance $F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_l:U^l\times [0, 1] \to \mathbb R^m \times [0, 1]$ recursively. Take any concordance $F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$$F_1:U^1\times [0, 1] \to \mathbb R^m \times [0, 1]$ between $f_0|_{U^1}$$f_0|_{U^1}$ and $f_1|_{U^1}$$f_1|_{U^1}$. Let us define a concordance $F_l$$F_l$ between $f_0|_{U^l}$$f_0|_{U^l}$ and $f_1|_{U^l}$$f_1|_{U^l}$ using a concordance $F_{l-1}$$F_{l-1}$ of $U^{l-1}$$U^{l-1}$ between $f_0|_{U^{l-1}}$$f_0|_{U^{l-1}}$ and $f_1|_{U^{l-1}}$$f_1|_{U^{l-1}}$. Denote by $\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$$\phi:\partial D^i\times D^{n-i}\to \partial U^{l-1}$ the $l$$l$-th attaching map. Since $i\leq n-1$$i\leq n-1$, by Lemma 3.4.(a) it follows that there is an extension of $F_{l-1}$$F_{l-1}$ to a concordance

$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$ $$\displaystyle F_{l}:(U^{l-1}\cup_\phi D^i\times D^{n-i})\times [0, 1]\to\mathbb{R}^m\times [0, 1]$$

between the restriction of $f_0$$f_0$ and $f_1$$f_1$ to $U^{l-1}\cup_\phi D^i\times D^{n-i}$$U^{l-1}\cup_\phi D^i\times D^{n-i}$.

$\square$$\square$

Proof of Lemma 3.4.(a) using Lemma 3.4.(b). In the following text we identify $D^i\times D^{n-i}\times [0, 1]$$D^i\times D^{n-i}\times [0, 1]$ and $D^i\times [0, 1]\times D^{n-i}$$D^i\times [0, 1]\times D^{n-i}$. Define map

$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$ $$\displaystyle \bar{\phi}:\partial (D^i\times [0, 1])\times D^{n-i}\to \partial U\times[0, 1]\cup_{\phi\times 0} D^i\times D^{n-i}\times 0 \cup_{\phi\times 1} D^i\times D^{n-i}\times 1$$

by the formula:

Tex syntax error

$$\displaystyle \bar{\phi}(x, t, y)=\left\{\begin{array}{c} (x, y, t), & x\in D^i,\ t\in \{0, 1\},\ y\in D^{n-i} \\ (\phi(x, y), t), & x\in \partial D^i,\ t\in (0, 1),\ y\in D^{n-i}\end{array}\right.$$

Since

$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$ $$\displaystyle \dim(D^i\times 0\times [0, 1]) + \dim (U\times [0, 1])\leq (i+1)+(n+1)\text{ and } 2\dim(D^i\times 0\times [0, 1])+1\leq 2n+1\leq m+1,$$

by general position there is an embedding

$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$ $$\displaystyle G: D^i\times 0\times [0, 1]\to \mathbb R^m\times [0, 1]$$

such that $F\bar{\phi} = G$$F\bar{\phi} = G$ on $\partial (D^i \times 0\times [0, 1])$$\partial (D^i \times 0\times [0, 1])$ and $F(\mbox{Int} (U\times [0, 1]))$$F(\mbox{Int} (U\times [0, 1]))$ has a finite number of intersection points with $G(\mbox{Int}( D^i\times 0\times [0, 1]))$$G(\mbox{Int}( D^i\times 0\times [0, 1]))$. Then by an isotopy $G_t$$G_t$, where $G_0=G$$G_0=G$, fixed on $\partial (D^i\times 0\times [0, 1])$$\partial (D^i\times 0\times [0, 1])$ we can "push out" the self-intersection points toward $F(\partial (U\times [0, 1]))$$F(\partial (U\times [0, 1]))$ so that $G_1(D^i\times 0\times [0, 1])$$G_1(D^i\times 0\times [0, 1])$ does not intersect $G(U\times [0, 1])$$G(U\times [0, 1])$. Then we may assume that $F\cup G$$F\cup G$ is an concordance between the restrictions of $f_0$$f_0$ and $f_1$$f_1$ on $U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$$U\cup_{\phi|_{\partial D^i\times 0}} D^i\times 0$.

Denote by $A$$A$ the $(n-i)\times (n+1)$$(n-i)\times (n+1)$ matrix whose rightmost $(n-i)\times (n-i)$$(n-i)\times (n-i)$ submatrix is the identity matrix, and whose other elements are zeroes. Denote by $v$$v$ the field of $n-i$$n-i$ normal vectors on $\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$$\partial (D^i\times 0\times [0, 1])\subset D^i\times D^{n-i}\times [0, 1]$ whose $k$$k$-th vector has coordinates equal to the $k$$k$-th row in $A$$A$. Then $d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$$d\bar{\phi} (v)= (d\bar{\phi} (v_1),\ldots , d\bar{\phi} (v_{n-i}))$ is the vector field tangent to $\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$$\bar{\phi}(\partial (D^i\times [0, 1])\times D^{n-i})$. For $x\in G(\partial (D^i\times 0\times [0, 1])$$x\in G(\partial (D^i\times 0\times [0, 1])$ denote by $v'$$v'$ the projection of $d(F\cup f_0\cup f_1)d\phi (v(x))$$d(F\cup f_0\cup f_1)d\phi (v(x))$ to the intersection of normal space to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$, and tangent space to $F(\partial(U\times [0, 1]))$$F(\partial(U\times [0, 1]))$. Since $i<m+1-(n-i)$$i<m+1-(n-i)$, it follows that $\pi_{i}(V_{m+1, n-i})=0$$\pi_{i}(V_{m+1, n-i})=0$. Hence there is an extension of $v'$$v'$ to a linear independent field of vectors normal to $G(D^i\times 0\times [0, 1])$$G(D^i\times 0\times [0, 1])$. Then by Lemma 3.4.(b) there is an extension of $F\cup G$$F\cup G$ to a concordance $(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$$(U \cup_{\phi} D^i\times D^{n-i})\times [0, 1]$.

$\square$$\square$

4 Example of non-isotopic embeddings

The following example is folklore.

Example 4.1. Let $N=S^k\times [0, 1]$$N=S^k\times [0, 1]$ be the cylinder over $S^k$$S^k$.

(a) Then there exist non-isotopic embeddings of $N$$N$ into $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$.

(b) Then for each $a\in\mathbb Z$$a\in\mathbb Z$ there exist an embedding $f\colon N\to\mathbb R^{2k+1}$$f\colon N\to\mathbb R^{2k+1}$ such that $\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$$\mathrm{lk}(f(S^k\times 0), f(S^k\times 1))=a$.

(c) Then $l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$$l\colon \mathrm{Emb}^{2k+1}N\to\mathbb Z$ defined by the formula $l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$$l([f]) = \mathrm{lk}(f(S^k\times 0), f(S^k\times 1))$ is well-defined and is a bijection for $k\geqslant2$$k\geqslant2$.

Proof. Proof of part (b). Informally speaking by twisting a ribbon one can obtain arbitrary value of linking coefficient. Let $h\colon S^k\to S^k$$h\colon S^k\to S^k$ be a map of degree $a$$a$. (To prove part (a) it is sufficient to take as $h$$h$ the identity map of $S^k$$S^k$ as a map of degree one and the constant map as a map of degree zero.) Define $g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$$g\colon S^k\times [0, 1] \to D^{k+1}\times S^k$ by the formula $g(x, t) = (x, h(x)t)$$g(x, t) = (x, h(x)t)$.

Let $f=\mathrm i\circ g$$f=\mathrm i\circ g$, where $\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$$\mathrm i = \mathrm i_{2k+1, k}\colon D^{k+1}\times S^k \to \mathbb R^{2k+1}$ is the standard embedding.Thus $\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$$\mathrm{lk}(f(S^k\times0), f(S^k\times1)) = a$.

Proof of part (c). Clearly $l$$l$ is well-defined. By (b) $l$$l$ is surjective. Now take any two embeddings $f_1, f_2$$f_1, f_2$ such that $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$. Each embedding of a cylinder gives an embedding of a sphere with a normal field. Moreover, isotopic embeddings of cylinders gives isotopic embeddings of spheres with normal fields.

Since $k\geqslant 2$$k\geqslant 2$ Unknotting Spheres Theorem implies that there exists an isotopy of $f_1|_{S^k\times 0}$$f_1|_{S^k\times 0}$ and $f_2|_{S^k\times 0}$$f_2|_{S^k\times 0}$. Thus we can assume $f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$$f_1|_{S^k\times 0} = f_2|_{S^k\times 0}$. Since $l([f_1]) = l([f_2])$$l([f_1]) = l([f_2])$ it follows that normal fields on $f_1(S^k\times 0)$$f_1(S^k\times 0)$ and $f_2(S^k\times 0)$$f_2(S^k\times 0)$ are homotopic in class of normal fields. This implies $f_1$$f_1$ and $f_2$$f_2$ are isotopic.

$\square$$\square$

(a): Embeddings $f_1$$f_1$ (top) and $f_2$$f_2$ (bottom); (b): the vector field depicts the difference $s_i-f_i$$s_i-f_i$, $i=1,2$$i=1,2$, so the ends of the vector field define the section $s_i$$s_i$; (c): embedding $s_ix\sqcup f_iy$$s_ix\sqcup f_iy$; (d): embedding $s_iy\sqcup f_ix$$s_iy\sqcup f_ix$.

Denote $1_k:=(1,0,\ldots,0)\in S^k$$1_k:=(1,0,\ldots,0)\in S^k$.

Example 4.2. Let $N=S^k\times S^1$$N=S^k\times S^1$. Assume $k>2$$k>2$. Then there exists a bijection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$$l([f])=\mathrm{lk}(S^k\times1_1, S^k\times-1_1)$.

The surjectivity of $l$$l$ is given analogously to Proposition 4.1(b). The injectivity of $l$$l$ follows from forgetful bijection $\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$$\mathrm{Emb}^{2k+1}N_0\to\mathrm{Emb}^{2k+1}S_k\times[0,1]$ between embeddings of $N_0$$N_0$ and a cylinder.

This example shows that Theorem 7.4 fails for $k=0$$k=0$.

Example 4.3. Let $N=S^k_a\times S^1 \# S^k_b\times S^1$$N=S^k_a\times S^1 \# S^k_b\times S^1$ be the connected sum of two tori. Then there exists a surjection $l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$$l\colon\mathrm{Emb}^{2k+1}N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$$l([f])=\mathrm{lk}(f(S^k_a\times1_1), f(S^k_b\times1_1))$.

To prove the surjectivity of $l$$l$ it is sufficient to take linked $k$$k$-spheres in $\mathbb R^{2k+1}$$\mathbb R^{2k+1}$ and consider an embedded boundary connected sum of ribbons containing these two spheres.

Example 4.4. (a) Let $N_0$$N_0$ be the punctured 2-torus containing the meridian $x$$x$ and the parallel $y$$y$ of the torus. For each embedding $f\colon N_0\to\mathbb R^3$$f\colon N_0\to\mathbb R^3$ denote by $s$$s$ the normal field of $\epsilon$$\epsilon$-length vectors to $f(N_0)$$f(N_0)$ defined by orientation on $N_0$$N_0$ (see figure (b)). Then there exists a surjection $l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$$l\colon\mathrm{Emb}^3 N_0\to\mathbb Z$ defined by the formula $l([f])=\mathrm{lk}(f(x), s(y))$$l([f])=\mathrm{lk}(f(x), s(y))$.

(b) Let $f_1,f_2\colon N_0\to\R^3$$f_1,f_2\colon N_0\to\R^3$ be two embeddings shown on figure (a). Figure (c) shows that $l(f_1)=1$$l(f_1)=1$ and $l(f_2)=0$$l(f_2)=0$ which proves the intuitive fact that $f_1$$f_1$ and $f_2$$f_2$ are not isotopic. (Notice that the restrictions of $f_1$$f_1$ and $f_2$$f_2$ on $x\cup y$$x\cup y$ are isotopic!) If we use the opposite normal vector field $s'=-s$$s'=-s$, the values of $l(f_1)$$l(f_1)$ and $l(f_2)$$l(f_2)$ will change but will still be different (see figure (d)).

5 Seifert linking form

In this section assume that

• $N$$N$ is any closed orientable connected $n$$n$-manifold,

• $f\colon N_0 \to \mathbb R^{2n-1}$$f\colon N_0 \to \mathbb R^{2n-1}$ is any embedding,

• if the (co)homology coefficients are omitted, then they are $\mathbb Z$$\mathbb Z$,

• $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion free (these two assumptions are not required in Lemma 5.4).

By $N_0$$N_0$ we denote the closure of the complement in $N$$N$ to an closed $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

The following folklore result holds.

Lemma 5.1. There exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

Proof. There is an obstruction (Euler class) $\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$$\bar e=\bar e(f)\in H^{n-1}(N_0)\cong H_1(N_0, \partial N_0)\cong H_1(N)$ to existence of a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

A normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$ has dimension $n-1$$n-1$. As $n$$n$ is even thus $n-1$$n-1$ is odd. Thus if we replace a general position normal field by its opposite then the obstruction will change sign. Therefore $\bar e=-\bar e$$\bar e=-\bar e$. Since $H_1(N)$$H_1(N)$ is torsion free, it follows that $\bar e=0$$\bar e=0$.

Since $N_0$$N_0$ has non-empty boundary, we have that $N_0$$N_0$ is homotopy equivalent to an $(n-1)$$(n-1)$-complex. The dimension of this complex equals the dimension of normal space to $f(N_0)$$f(N_0)$ at any point of $f(N_0)$$f(N_0)$. Since $\bar e=0$$\bar e=0$, it follows that there exists a nowhere vanishing normal vector field to $f(N_0)$$f(N_0)$.

$\square$$\square$

Denote by $x, y$$x, y$ two disjoint $(n-1)$$(n-1)$-cycles in $N_0$$N_0$ with integer coefficients.

Definition 5.2. Denote

$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$ $$\displaystyle L(f)(x,y) = \mathrm{lk}(f(x), s(y)) + \mathrm{lk}(s(x), f(y)),$$

where $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$ and $s(x), s(y)$$s(x), s(y)$ are the results of the shift of $f(x), f(y)$$f(x), f(y)$ by $s$$s$.

Lemma 5.3 ($L$$L$ is well-defined). The integer $L(f)(x, y)$$L(f)(x, y)$:

• is well-defined, i.e. does not change when $s$$s$ is replaced by $s'$$s'$,
• does not change when $x$$x$ or $y$$y$ are changed to homologous cycles and,
• does not change when $f$$f$ is changed to an isotopic embedding.

The first bullet was stated and proved in unpublished update of [Tonkonog2010], other two bullets are simple.

Lemma 5.4. Let $s,s'$$s,s'$ be two nowhere vanishing normal vector fields to $f(N_0)$$f(N_0)$. Then

$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$ $$\displaystyle \mathrm{lk}(f(x),s(y))-\mathrm{lk}(f(x),s'(y))=d(s,s')\cap x\cap y$$

where $s(y)$$s(y)$ is the result of the shift of $f(y)$$f(y)$ by $s$$s$, and $d(s,s')\in H_2(N_0)$$d(s,s')\in H_2(N_0)$ is (Poincare dual to) the first obstruction to $s,s'$$s,s'$ being homotopic in the class of the nowhere vanishing vector fields.

This Lemma is proved in [Saeki1999, Lemma 2.2] for $n=3$$n=3$, but the proof is valid in all dimensions.

Proof of Lemma 5.3. The first bullet point follows because:

$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$ $$\displaystyle \begin{aligned} \mathrm{lk}(f(x),s(y))+&\mathrm{lk}(s(x),f(y))&-\mathrm{lk}(f(x),s'(y))-&\mathrm{lk}(s'(x),f(y))= \\ \mathrm{lk}(f(x),s(y))+&(-1)^n\,\mathrm{lk}(f(y),s(x))&-\mathrm{lk}(f(x),s'(y))-&(-1)^n\,\mathrm{lk}(f(y),s'(x))=\\ &&d(s,s')\cap x\cap y+&(-1)^n \,d(s,s')\cap y\cap x=\\ &&d(s,s')\cap x\cap y(1+&(-1)^n(-1)^{n-1})=0. \end{aligned}$$

Here the second equality follows from Lemma 5.4.

For each two homologous $(n-1)$$(n-1)$-cycles $x, x'$$x, x'$ in $N_0$$N_0$, the image of the homology between $x$$x$ and $x'$$x'$ is a $n$$n$-chain $X$$X$ of $f(N_0)$$f(N_0)$ such that $\partial X = f(x) - f(x')$$\partial X = f(x) - f(x')$. Since $s$$s$ is a nowhere vanishing normal field to $f(N_0)$$f(N_0)$, this implies that the supports of $s(y)$$s(y)$ and $X$$X$ are disjoint. Hence $\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$$\mathrm{lk}(f(x), s(y)) = \mathrm{lk}(f(x'), s(y))$.

Since isotopy of $f$$f$ is a map from $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$ to $\mathbb R^{2n-1}\times [0, 1]$$\mathbb R^{2n-1}\times [0, 1]$, it follows that this isotopy gives an isotopy of the link $f(x)\sqcup s(y)$$f(x)\sqcup s(y)$. Now the third bullet point follows because the linking coefficient is preserved under isotopy.

$\square$$\square$

Lemma 5.3 implies that $L(f)$$L(f)$ generates a bilinear form $H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$$H_{n-1}(N_0)\times H_{n-1}(N_0)\to\mathbb Z$ denoted by the same letter.

Denote by $\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$$\rho_2 \colon H_*(N)\to H_*(N;\mathbb Z_2)$ the reduction modulo $2$$2$.

Define the dual to Stiefel-Whitney class $\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$$\mathrm{PD}\bar w_{n-2}(N_0)\in H_2(N_0, \partial N_0; \mathbb Z_2)$ to be the class of the cycle on which two general position normal fields to $f(N_0)$$f(N_0)$ are linearly dependent.

Lemma 5.5. For every $X, Y \in H_{n-1}(N_0)$$X, Y \in H_{n-1}(N_0)$ the following equality holds:

$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$ $$\displaystyle \rho_2L(f)(X, Y) = \mathrm{PD}\bar w_{n-2}(N_0)\cap\rho_2X\cap\rho_2Y.$$

This Lemma was stated in a unpublished update of [Tonkonog2010], the following proof is presented in [Fedorov2021] using the idea from that update. See also an analogous lemma for closed manifolds in [Crowley&Skopenkov2016, Lemma 2.2].

Proof of Lemma 5.5. Let $-s$$-s$ be the normal field to $f(N_0)$$f(N_0)$ opposite to $s$$s$. We get

$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$ $$\displaystyle \begin{aligned} L(f)(X, Y) &\underset{2}\equiv \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(s(X), f(Y)) = \\ &= \mathrm{lk}(f(X), s(Y)) - \mathrm{lk}(f(X), -s(Y)) = \\ &= d(s, -s)\cap X\cap Y . \end{aligned}$$

The first congruence is clear.

The second equality holds because if we shift the link $s(X)\sqcup f(Y)$$s(X)\sqcup f(Y)$ by $-s$$-s$, we get the link $f(X)\sqcup -s(Y)$$f(X)\sqcup -s(Y)$ and the linking coefficient will not change after this shift.

The third equality follows from Lemma 5.4.

Thus it is sufficient to show that $\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$$\rho_2d(s, -s) = \mathrm{PD}\bar w_{n-2}(N_0)$. Denote by $s'$$s'$ a general perturbation of $s$$s$. We get:

$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$ $$\displaystyle \rho_2 d(s, -s) = \rho_2 d(s', -s) = \mathrm{PD}\bar w_{n-2}(N_0).$$

The first equality holds because $s'$$s'$ and $s$$s$ are homotopic in the class of nowhere vanishing normal vector fields. Let us prove the second equality. The linear homotopy between $s'$$s'$ and $-s$$-s$ degenerates only at those points $x$$x$ where $s'(x)=s(x)$$s'(x)=s(x)$. These points $x$$x$ are exactly points where $s'(x)$$s'(x)$ and $s(x)$$s(x)$ are linearly dependent. All those point $x$$x$ form a $2$$2$-cycle modulo two in $N_0$$N_0$. The homotopy class of this $2$$2$-cycle is $\mathrm{PD}\bar w_{n-2}(N_0)$$\mathrm{PD}\bar w_{n-2}(N_0)$ by the definition of Stiefel-Whitney class.

$\square$$\square$

6 Classification theorems

Here we state classification results that are neither unknotting nor embeddability theorems for manifolds with boundary.

Let $N$$N$ be a closed orientable connected $n$$n$-manifold. By $N_0$$N_0$ we denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere. For a free Abelian group $A$$A$, let $B_n^∗A$$B_n^∗A$ be the group of bilinear forms $\phi \colon A \times A \to \mathbb Z$$\phi \colon A \times A \to \mathbb Z$ such that $\phi(x, y) = (−1)^n \phi(y, x)$$\phi(x, y) = (−1)^n \phi(y, x)$ and $\phi(x, x)$$\phi(x, x)$ is even for each $x$$x$ (the second condition automatically holds for n odd).

Definition 6.1. For each even $n$$n$ define an invariant $W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$$W\Lambda\colon\mathrm{Emb}^{2n-1}N_0\to H_1(N;\mathbb Z_2)$. For each embedding $f\colon N_0\to\mathbb R^{2n-1}$$f\colon N_0\to\mathbb R^{2n-1}$ construct any PL embedding $g\colon N\to\mathbb R^{2n}$$g\colon N\to\mathbb R^{2n}$ by adding a cone over $f(\partial N_0)$$f(\partial N_0)$. Now let $W\Lambda([f]) = W(g)$$W\Lambda([f]) = W(g)$, where $W$$W$ is Whitney invariant, [Skopenkov2016e, $\S$$\S$5].

Lemma 6.2. The invariant $W\Lambda$$W\Lambda$ is well-defined for $n\ge4$$n\ge4$.

Proof. Note that Unknotting Spheres Theorem implies that $\partial N_0$$\partial N_0$ unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus $f|_{\partial N_0}$$f|_{\partial N_0}$ can be extended to embedding of an $n$$n$-ball $B^n$$B^n$ into $\mathbb R^{2n}$$\mathbb R^{2n}$. Unknotting Spheres Theorem implies that $n$$n$-sphere unknots in $\mathbb R^{2n}$$\mathbb R^{2n}$. Thus all extensions of $f$$f$ are isotopic in PL category. Note also that if $f$$f$ and $g$$g$ are isotopic then their extensions are isotopic as well. And Whitney invariant $W$$W$ is invariant for PL embeddings.

$\square$$\square$

Definition 6.3 of $G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$G:\mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ if $n$$n$ is even and $H_1(N)$$H_1(N)$ is torsion-free. Take a collection $\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$$\{f_z\}_{z\in H_1(N;\Z_{(n-1)})}\subset \mathrm{Emb}^{2n-1} N_0$ such that $W\Lambda(f_z)=z$$W\Lambda(f_z)=z$. For each $f$$f$ such that $W\Lambda(f)=z$$W\Lambda(f)=z$ define

$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$ $$\displaystyle G(f)(x,y):=\frac{1}{2}\left(L(f)(x,y)-L(f_z)(x,y)\right)$$

where $x,y\in H_{n-1}(N_0)$$x,y\in H_{n-1}(N_0)$.

Note also that $G$$G$ depends on choice of collection $\{f_z\}$$\{f_z\}$. The following Theorems hold for any choice of $\{f_z\}$$\{f_z\}$.

Theorem 6.4. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. The map

$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$ $$\displaystyle G\times W\Lambda:\ \mathrm{Emb}^{2n-1}N_0\to B_n^* H_{n-1}(N) \times H_1(N;\Z_{2}),$$

is one-to-one.

Lemma 6.5. For each even $n\in H_{n-1}(N)$$n\in H_{n-1}(N)$ and each $x$$x$ the following equality holds: $W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$$W\Lambda(f)\cap \rho_2(x) = \rho_2\left(\frac{1}{2}L(f)(x, x) - \frac{1}{2}L(f_0)(x, x)\right)$.

An equivalemt statement of Theorem 6.4:

Theorem 6.6. Let $N$$N$ be a closed connected orientable $n$$n$-manifold with $H_1(N)$$H_1(N)$ torsion-free, $n\ge 4$$n\ge 4$, $n$$n$ even. Then

(a) The map $L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$$L: \mathrm{Emb}^{2n-1}N_0\to B_n^*H_{n-1}(N)$ is an injection.

(b) The image of $L$$L$ consists of all symmetric bilinear forms $\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$$\phi:H_{n-1}(N)\times H_{n-1}(N)\to \Z$ such that $\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$$\rho_2\phi(x,y)= \bar w_2(N_0)\cap\rho_2(x\cap y)$. Here $\bar w_2(N_0)$$\bar w_2(N_0)$ is the normal Stiefel-Whitney class.

This is the main Theorem of [Tonkonog2010]

7 A generalization to highly-connected manifolds

For simplicity in this paragraph we consider only punctured manifolds, see $\S$$\S$8 for a generalization.

Denote by $N$$N$ a closed $n$$n$-manifold. By $N_0$$N_0$ denote the complement in $N$$N$ to an open $n$$n$-ball. Thus $\partial N_0$$\partial N_0$ is the $(n-1)$$(n-1)$-sphere.

Theorem 7.1. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $n\geq 2k+3$$n\geq 2k+3$, then $N$$N$ embeds into $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$.

(b) If $n\geq 2k+2$$n\geq 2k+2$ and $k\geq0$$k\geq0$, then $N_0$$N_0$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

Part (a) is proved in [Haefliger1961, Existence Theorem (a)] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.1], [Irwin1965, Corollary 1.3] for PL case.

Part (b) is proved in [Hirsch1961a, Corollary 4.2] for the Diff case and in [Penrose&Whitehead&Zeeman1961, Theorem 1.2] for the PL case.

Theorem 7.2. Assume that $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold.

(a) If $m \ge 2n - k + 1$$m \ge 2n - k + 1$ and $n\ge2k + 2$$n\ge2k + 2$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

(b) If $m \ge 2n - k$$m \ge 2n - k$ and $n\ge k + 3$$n\ge k + 3$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$ then any two embeddings of $N_0$$N_0$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Part (a) is Theorem 2.4 of the survey [Skopenkov2016c, $\S$$\S$ 2], and is proved in [Zeeman1963, Corollary 2 of Theorem 24 in Chapter 8] and [Haefliger1961, Existence Theorem (b) in p. 47].

Part (b) is proved in [Hudson1969, Theorem 10.3] for the PL case, using concordance implies isotopy theorem.

For $k>1$$k>1$ part (b) is a corollary of Theorem 7.4 below. For $k=0$$k=0$ part (b) coincides with Theorem 2.2b.

Proof of Theorem 7.2(b) for $k=1$$k=1$. By Theorem 9.6 below every two immersions of $N_0$$N_0$ into $\mathbb R^{2n-1}$$\mathbb R^{2n-1}$ are regulary homotopic. Hence for every two embeddings $f,g\colon N_0\to\mathbb R^{2n-1}$$f,g\colon N_0\to\mathbb R^{2n-1}$ there exist an immersion $F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$$F\colon N_0\times[0,1]\to\mathbb R^{2n-1}\times[0,1]$ such that $F(x, 0) = (f(x), 0)$$F(x, 0) = (f(x), 0)$ and $F(x, 1)=(g(x), 1)$$F(x, 1)=(g(x), 1)$ for each $x\in N_0$$x\in N_0$. It follows from Theorem 8.3 that $N_0$$N_0$ collapses to an $(n-2)$$(n-2)$-dimensional subcomplex $X\subset N_0$$X\subset N_0$ of some triangulation of $N_0$$N_0$. By general position we may assume that $F|_{X\times[0,1]}$$F|_{X\times[0,1]}$ is an embedding, because $2(n-1) < 2n$$2(n-1) < 2n$. Since $F$$F$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N_0\to M$$h\colon N_0\to M$. It is clear that $f$$f$ is isotopic to $f\circ h$$f\circ h$ and $g$$g$ is isotopic to $g\circ h$$g\circ h$. Thus, the restriction $F|_{M\times[0,1]}$$F|_{M\times[0,1]}$ is a concordance of $f\circ h$$f\circ h$ and $g\circ h$$g\circ h$. By concordance implies isotopy Theorem $f$$f$ and $g$$g$ are isotopic.

$\square$$\square$

Conjecture 7.3. Assume that $N$$N$ is a closed $1$$1$-connected $4$$4$-manifold. Then any two embeddings of $N_0$$N_0$ in $\mathbb R^7$$\mathbb R^7$ are isotopic.

We may hope to get around the restrictions of Theorem 8.3 using the deleted product criterion.

Theorem 7.4. Assume $N$$N$ is a closed $k$$k$-connected $n$$n$-manifold. Then for each $k\ge1$$k\ge1$ there exists a bijection

$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$ $$\displaystyle W_0'\colon \mathrm{Emb}^{2n-k-1}(N_0)\to H_{k+1}(N;\mathbb Z_{(n-k-1)}),$$

where $\mathbb Z_{(s)}$$\mathbb Z_{(s)}$ denote $\mathbb Z$$\mathbb Z$ for $s$$s$ even and $\mathbb Z_2$$\mathbb Z_2$ for $s$$s$ odd.

For definition of $W_0'$$W_0'$ and the proof of the latter Theorem see [Skopenkov2010, Lemma 2.2($W_0'$$W_0'$)]. The latter theorem was essentially proven in [Vrabec1989, Theorem 2.1]. Latter Theorem is essentially known result. Compare to the Theorem 6.6, which describes $\mathrm{Emb}^{2n-1}(N_0)$$\mathrm{Emb}^{2n-1}(N_0)$ and differs from the general case.

8 Comments on non-spherical boundary

Theorem 8.1. Assume that $N$$N$ is a compact $k$$k$-connected $n$$n$-manifold, $\partial N\neq\emptyset$$\partial N\neq\emptyset$, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $k< n - 3$$k< n - 3$. Then $N$$N$ embeds into $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$.

This is [Wall1965, Theorem on p.567].

Proof. By Theorem 9.5 below there exists an immersion $f\colon N\to\mathbb R^{2n-k-1}$$f\colon N\to\mathbb R^{2n-k-1}$. Since $N$$N$ is $k$$k$-connected it follows from Theorem 8.3 that $N$$N$ collapses to an $(n-k-1)$$(n-k-1)$-dimensional subcomplex $X\subset N$$X\subset N$ of some triangulation of $N$$N$. By general position we may assume that $f|_X$$f|_X$ is an embedding, because $2(n-k) < 2n-k-1$$2(n-k) < 2n-k-1$. Since $f$$f$ is an immersion, it follows that $X$$X$ has a sufficiently small regular neighbourhood $M\supset X$$M\supset X$ such that $f|_{M}$$f|_{M}$ is an embedding. Since regular neighbourhood is unique up to homeomorphism, there exists a homeomorphism $h\colon N\to M$$h\colon N\to M$. It is clear that $f\circ h$$f\circ h$ is an embedding.

$\square$$\square$

Theorem 8.2. Assume that $N$$N$ is a $n$$n$-manifold. If $N$$N$ has $(n-k-1)$$(n-k-1)$-dimensional spine, $\partial N \neq \emptyset$$\partial N \neq \emptyset$, $m \ge 2n - k$$m \ge 2n - k$, then any two embeddings of $N$$N$ into $\mathbb R^m$$\mathbb R^m$ are isotopic.

Proof is similar to the proof of theorem 7.2.

For a compact connected $n$$n$-manifold with boundary, the property of having an $(n − k − 1)$$(n − k − 1)$-dimensional spine is close to $k$$k$-connectedness. Indeed, the following theorem holds.

Theorem 8.3. Every compact connected $n$$n$-manifold $N$$N$ with boundary for which $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected, $\pi_1(\partial N)=0$$\pi_1(\partial N)=0$, $k + 3 \le n$$k + 3 \le n$ and $(n, k) \notin \{(5, 2), (4, 1)\}$$(n, k) \notin \{(5, 2), (4, 1)\}$, has an $(n − k − 1)$$(n − k − 1)$-dimensional spine.

For this result see [Wall1964a, Theorem 5.5] and [Horvatic1969, Lemma 5.1 and Remark 5.2]. See also valuable remarks in [Levine&Lidman2018] and [Skopenkov2019].

9 Comments on immersions

Theorem 9.1.[Smale-Hirsch; [Hirsch1959] and [Haefliger&Poenaru1964]] The space of immersions of a manifold in $\R^m$$\R^m$ is homotopy equivalent to the space of linear monomorphisms from $TM$$TM$ to $\R^m$$\R^m$.

Theorem 9.2.[[Hirsch1959, Theorem 6.4]] If $N$$N$ is immersible in $\R^{m+r}$$\R^{m+r}$ with a normal $r$$r$-field, then $N$$N$ is immersible in $\R^m$$\R^m$.

Theorem 9.3. Every $n$$n$-manifold $N$$N$ with non-empty boundary is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Theorem 9.4.[Whitney; [Hirsch1961a, Theorem 6.6]] Every $n$$n$-manifold $N$$N$ is immersible in $\R^{2n-1}$$\R^{2n-1}$.

Denote by $V_{m,n}$$V_{m,n}$ is Stiefel manifold of $n$$n$-frames in $\R^m$$\R^m$.

Theorem 9.5. Suppose $N$$N$ is a $n$$n$-manifold with non-empty boudary, $(N,\partial N)$$(N,\partial N)$ is $k$$k$-connected. Then $N$$N$ is immersible in $\R^m$$\R^m$ for each $m\geq2n-k-1$$m\geq2n-k-1$.

Proof. It suffices to show that exists an immersion of $N$$N$ in $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. It suffices to show that exists a linear monomorphism from $TM$$TM$ to $\mathbb R^{2n-k-1}$$\mathbb R^{2n-k-1}$. Let us construct such a linear monomorphism by skeleta of $N$$N$. It is clear that a linear monomorphism exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the linear monomorphism from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$$H_{n-r}(N, \partial N; \pi_{r-1}(V_{2n-k-1,n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_{r-1}(V_{2n-k-1, n}) = 0$$\pi_{r-1}(V_{2n-k-1, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and has non-empty boundary.

Thus the obstruction is always zero and such linear monomorphism exists.

$\square$$\square$

Theorem 9.6. Suppose $N$$N$ is a connected $n$$n$-manifold with non-empty boudary, $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $m\geq2n-k$$m\geq2n-k$. Then every two immersions of $N$$N$ in $\R^m$$\R^m$ are regulary homotopic.

Proof. It suffies to show that exists homomotphism of any two linear monomorphisms from $TM$$TM$ to $\mathbb R^{2n-k}$$\mathbb R^{2n-k}$. Lets cunstruct such homotopy on each $r$$r$-skeleton of $N$$N$. It is clear that homotopy exists on $0$$0$-skeleton of $N$$N$.

The obstruction to extend the homotopy from $(r-1)$$(r-1)$-skeleton to $r$$r$-skeleton lies in $H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$$H_{n-r}(N, \partial N; \pi_r(V_{2n-k, n}))$.

For $r=1,\ldots,n-k-1$$r=1,\ldots,n-k-1$ we know $\pi_r(V_{2n-k, n}) = 0$$\pi_r(V_{2n-k, n}) = 0$. For $r=n-k,\ldots, n$$r=n-k,\ldots, n$ we have $H_{n-r}(N, \partial N; *) = 0$$H_{n-r}(N, \partial N; *) = 0$ since $(N, \partial N)$$(N, \partial N)$ is $k$$k$-connected and $N$$N$ has non-empty boundary.

Thus the obstruction is always zero and such homotopy of linear monomorphisms exists.

$\square$$\square$

10 References

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